diff --git a/sklearn/_config.py b/sklearn/_config.py
index 05549c88a9ddc..af1342d69c5b4 100644
--- a/sklearn/_config.py
+++ b/sklearn/_config.py
@@ -8,18 +8,19 @@
 from contextlib import contextmanager as contextmanager
 
 _global_config = {
+    "array_api_dispatch": False,
     "assume_finite": bool(os.environ.get("SKLEARN_ASSUME_FINITE", False)),
-    "working_memory": int(os.environ.get("SKLEARN_WORKING_MEMORY", 1024)),
-    "print_changed_only": True,
     "display": "diagram",
+    "enable_cython_pairwise_dist": True,
+    "enable_metadata_routing": False,
+    "inverse_transform_output": "default",
     "pairwise_dist_chunk_size": int(
         os.environ.get("SKLEARN_PAIRWISE_DIST_CHUNK_SIZE", 256)
     ),
-    "enable_cython_pairwise_dist": True,
-    "array_api_dispatch": False,
-    "transform_output": "default",
-    "enable_metadata_routing": False,
+    "print_changed_only": True,
     "skip_parameter_validation": False,
+    "transform_output": "default",
+    "working_memory": int(os.environ.get("SKLEARN_WORKING_MEMORY", 1024)),
 }
 _threadlocal = threading.local()
 
@@ -66,6 +67,7 @@ def set_config(
     enable_cython_pairwise_dist=None,
     array_api_dispatch=None,
     transform_output=None,
+    inverse_transform_output=None,
     enable_metadata_routing=None,
     skip_parameter_validation=None,
 ):
@@ -205,6 +207,8 @@ def set_config(
         local_config["array_api_dispatch"] = array_api_dispatch
     if transform_output is not None:
         local_config["transform_output"] = transform_output
+    if inverse_transform_output is not None:
+        local_config["inverse_transform_output"] = inverse_transform_output
     if enable_metadata_routing is not None:
         local_config["enable_metadata_routing"] = enable_metadata_routing
     if skip_parameter_validation is not None:
@@ -222,6 +226,7 @@ def config_context(
     enable_cython_pairwise_dist=None,
     array_api_dispatch=None,
     transform_output=None,
+    inverse_transform_output=None,
     enable_metadata_routing=None,
     skip_parameter_validation=None,
 ):
@@ -366,6 +371,7 @@ def config_context(
         enable_cython_pairwise_dist=enable_cython_pairwise_dist,
         array_api_dispatch=array_api_dispatch,
         transform_output=transform_output,
+        inverse_transform_output=inverse_transform_output,
         enable_metadata_routing=enable_metadata_routing,
         skip_parameter_validation=skip_parameter_validation,
     )
diff --git a/sklearn/cluster/_feature_agglomeration.py b/sklearn/cluster/_feature_agglomeration.py
index 1983aae00ecbb..1a59b91d1a391 100644
--- a/sklearn/cluster/_feature_agglomeration.py
+++ b/sklearn/cluster/_feature_agglomeration.py
@@ -19,7 +19,7 @@
 # Mixin class for feature agglomeration.
 
 
-class AgglomerationTransform(TransformerMixin):
+class AgglomerationTransform(TransformerMixin, auto_wrap_output_keys=("transform",)):
     """
     A class for feature agglomeration via the transform interface.
     """
@@ -84,6 +84,9 @@ def inverse_transform(self, X=None, *, Xt=None):
             A vector of size `n_samples` with the values of `Xred` assigned to
             each of the cluster of samples.
         """
+        # because this method takes X and Xt(deprecated),
+        # auto_wrap_output is not configured for this method.
+
         X = _deprecate_Xt_in_inverse_transform(X, Xt)
 
         check_is_fitted(self)
diff --git a/sklearn/decomposition/_nmf.py b/sklearn/decomposition/_nmf.py
index 6be97f2223fb5..21f02e90fbaf9 100644
--- a/sklearn/decomposition/_nmf.py
+++ b/sklearn/decomposition/_nmf.py
@@ -1132,7 +1132,13 @@ def non_negative_factorization(
     return W, H, n_iter
 
 
-class _BaseNMF(ClassNamePrefixFeaturesOutMixin, TransformerMixin, BaseEstimator, ABC):
+class _BaseNMF(
+    ClassNamePrefixFeaturesOutMixin,
+    TransformerMixin,
+    BaseEstimator,
+    ABC,
+    auto_wrap_output_keys=("transform",),
+):
     """Base class for NMF and MiniBatchNMF."""
 
     # This prevents ``set_split_inverse_transform`` to be generated for the
@@ -1318,6 +1324,8 @@ def inverse_transform(self, X=None, *, Xt=None):
             Returns a data matrix of the original shape.
         """
 
+        # because this method takes X and Xt(deprecated),
+        # auto_wrap_output is not configured for this method.
         X = _deprecate_Xt_in_inverse_transform(X, Xt)
 
         check_is_fitted(self)
diff --git a/sklearn/preprocessing/_discretization.py b/sklearn/preprocessing/_discretization.py
index 6a6a739c469fa..a6f9369e8ce23 100644
--- a/sklearn/preprocessing/_discretization.py
+++ b/sklearn/preprocessing/_discretization.py
@@ -22,7 +22,9 @@
 from ._encoders import OneHotEncoder
 
 
-class KBinsDiscretizer(TransformerMixin, BaseEstimator):
+class KBinsDiscretizer(
+    TransformerMixin, BaseEstimator, auto_wrap_output_keys=("transform",)
+):
     """
     Bin continuous data into intervals.
 
@@ -412,6 +414,9 @@ def inverse_transform(self, X=None, *, Xt=None):
         Xinv : ndarray, dtype={np.float32, np.float64}
             Data in the original feature space.
         """
+        # because this method takes X and Xt(deprecated),
+        # auto_wrap_output is not configured for this method.
+
         X = _deprecate_Xt_in_inverse_transform(X, Xt)
 
         check_is_fitted(self)
diff --git a/sklearn/tests/test_config.py b/sklearn/tests/test_config.py
index fbdb0e2884d32..7e859b0348664 100644
--- a/sklearn/tests/test_config.py
+++ b/sklearn/tests/test_config.py
@@ -20,6 +20,7 @@ def test_config_context():
         "pairwise_dist_chunk_size": 256,
         "enable_cython_pairwise_dist": True,
         "transform_output": "default",
+        "inverse_transform_output": "default",
         "enable_metadata_routing": False,
         "skip_parameter_validation": False,
     }
@@ -38,6 +39,7 @@ def test_config_context():
             "pairwise_dist_chunk_size": 256,
             "enable_cython_pairwise_dist": True,
             "transform_output": "default",
+            "inverse_transform_output": "default",
             "enable_metadata_routing": False,
             "skip_parameter_validation": False,
         }
@@ -73,6 +75,7 @@ def test_config_context():
         "pairwise_dist_chunk_size": 256,
         "enable_cython_pairwise_dist": True,
         "transform_output": "default",
+        "inverse_transform_output": "default",
         "enable_metadata_routing": False,
         "skip_parameter_validation": False,
     }
diff --git a/sklearn/utils/_set_output.py b/sklearn/utils/_set_output.py
index 963e5e5bf6d77..18059b493a726 100644
--- a/sklearn/utils/_set_output.py
+++ b/sklearn/utils/_set_output.py
@@ -341,27 +341,30 @@ def _auto_wrap_is_configured(estimator):
     is manually disabled.
     """
     auto_wrap_output_keys = getattr(estimator, "_sklearn_auto_wrap_output_keys", set())
-    return (
-        hasattr(estimator, "get_feature_names_out")
-        and "transform" in auto_wrap_output_keys
+    return hasattr(estimator, "get_feature_names_out") and (
+        "transform" in auto_wrap_output_keys
+        or "inverse_transform" in auto_wrap_output_keys
     )
 
 
 class _SetOutputMixin:
     """Mixin that dynamically wraps methods to return container based on config.
 
-    Currently `_SetOutputMixin` wraps `transform` and `fit_transform` and configures
-    it based on `set_output` of the global configuration.
+    Currently `_SetOutputMixin` wraps `transform`, `fit_transform` and
+    `inverse_transform` and configures it based on `set_output` of the global
+    configuration.
 
     `set_output` is only defined if `get_feature_names_out` is defined and
     `auto_wrap_output_keys` is the default value.
     """
 
-    def __init_subclass__(cls, auto_wrap_output_keys=("transform",), **kwargs):
+    def __init_subclass__(
+        cls, auto_wrap_output_keys=("transform", "inverse_transform"), **kwargs
+    ):
         super().__init_subclass__(**kwargs)
 
         # Dynamically wraps `transform` and `fit_transform` and configure it's
-        # output based on `set_output`.
+        # `inverse_transform` output based on `set_output`.
         if not (
             isinstance(auto_wrap_output_keys, tuple) or auto_wrap_output_keys is None
         ):
@@ -375,6 +378,7 @@ def __init_subclass__(cls, auto_wrap_output_keys=("transform",), **kwargs):
         method_to_key = {
             "transform": "transform",
             "fit_transform": "transform",
+            "inverse_transform": "inverse_transform",
         }
         cls._sklearn_auto_wrap_output_keys = set()
 
@@ -390,7 +394,7 @@ def __init_subclass__(cls, auto_wrap_output_keys=("transform",), **kwargs):
             setattr(cls, method, wrapped_method)
 
     @available_if(_auto_wrap_is_configured)
-    def set_output(self, *, transform=None):
+    def set_output(self, *, inverse_transform=None, transform=None):
         """Set output container.
 
         See :ref:`sphx_glr_auto_examples_miscellaneous_plot_set_output.py`
@@ -414,13 +418,14 @@ def set_output(self, *, transform=None):
         self : estimator instance
             Estimator instance.
         """
-        if transform is None:
-            return self
-
-        if not hasattr(self, "_sklearn_output_config"):
-            self._sklearn_output_config = {}
-
-        self._sklearn_output_config["transform"] = transform
+        if transform is not None or inverse_transform is not None:
+            if not hasattr(self, "_sklearn_output_config"):
+                self._sklearn_output_config = {}
+
+            if inverse_transform is not None:
+                self._sklearn_output_config["inverse_transform"] = inverse_transform
+            if transform is not None:
+                self._sklearn_output_config["transform"] = transform
         return self