diff --git a/doc/source/whatsnew/v2.1.0.rst b/doc/source/whatsnew/v2.1.0.rst
index 68159cd211a5e..40dfe0b33d66c 100644
--- a/doc/source/whatsnew/v2.1.0.rst
+++ b/doc/source/whatsnew/v2.1.0.rst
@@ -93,6 +93,7 @@ Other API changes
 Deprecations
 ~~~~~~~~~~~~
 - Deprecated accepting slices in :meth:`DataFrame.take`, call ``obj[slicer]`` or pass a sequence of integers instead (:issue:`51539`)
+- Deprecating pinning ``group.name`` to each group in :meth:`SeriesGroupBy.aggregate` aggregations; if your operation requires utilizing the groupby keys, iterate over the groupby object instead (:issue:`41090`)
 -
 
 .. ---------------------------------------------------------------------------
diff --git a/pandas/core/groupby/generic.py b/pandas/core/groupby/generic.py
index 60c43b6cf0ecd..dca2d5676f71d 100644
--- a/pandas/core/groupby/generic.py
+++ b/pandas/core/groupby/generic.py
@@ -23,6 +23,7 @@
     Union,
     cast,
 )
+import warnings
 
 import numpy as np
 
@@ -49,6 +50,7 @@
     Substitution,
     doc,
 )
+from pandas.util._exceptions import find_stack_level
 
 from pandas.core.dtypes.common import (
     ensure_int64,
@@ -270,6 +272,16 @@ def aggregate(self, func=None, *args, engine=None, engine_kwargs=None, **kwargs)
                 #  pinned in _python_agg_general, only in _aggregate_named
                 result = self._aggregate_named(func, *args, **kwargs)
 
+                warnings.warn(
+                    "Pinning the groupby key to each group in "
+                    f"{type(self).__name__}.agg is deprecated, and cases that "
+                    "relied on it will raise in a future version. "
+                    "If your operation requires utilizing the groupby keys, "
+                    "iterate over the groupby object instead.",
+                    FutureWarning,
+                    stacklevel=find_stack_level(),
+                )
+
                 # result is a dict whose keys are the elements of result_index
                 result = Series(result, index=self.grouper.result_index)
                 result = self._wrap_aggregated_output(result)
@@ -407,6 +419,7 @@ def _aggregate_named(self, func, *args, **kwargs):
         for name, group in self.grouper.get_iterator(
             self._selected_obj, axis=self.axis
         ):
+            # needed for pandas/tests/groupby/test_groupby.py::test_basic_aggregations
             object.__setattr__(group, "name", name)
 
             output = func(group, *args, **kwargs)
@@ -1537,6 +1550,7 @@ def _transform_general(self, func, *args, **kwargs):
         except StopIteration:
             pass
         else:
+            # 2023-02-27 No tests broken by disabling this pinning
             object.__setattr__(group, "name", name)
             try:
                 path, res = self._choose_path(fast_path, slow_path, group)
@@ -1552,6 +1566,7 @@ def _transform_general(self, func, *args, **kwargs):
         for name, group in gen:
             if group.size == 0:
                 continue
+            # 2023-02-27 No tests broken by disabling this pinning
             object.__setattr__(group, "name", name)
             res = path(group)
 
@@ -1721,6 +1736,8 @@ def filter(self, func, dropna: bool = True, *args, **kwargs):
         gen = self.grouper.get_iterator(obj, axis=self.axis)
 
         for name, group in gen:
+            # 2023-02-27 no tests are broken this pinning, but it is documented in the
+            #  docstring above.
             object.__setattr__(group, "name", name)
 
             res = func(group, *args, **kwargs)
diff --git a/pandas/core/groupby/ops.py b/pandas/core/groupby/ops.py
index 726d75d705344..9a06a3da28e15 100644
--- a/pandas/core/groupby/ops.py
+++ b/pandas/core/groupby/ops.py
@@ -760,6 +760,11 @@ def apply_groupwise(
         zipped = zip(group_keys, splitter)
 
         for key, group in zipped:
+            # Pinning name is needed for
+            #  test_group_apply_once_per_group,
+            #  test_inconsistent_return_type, test_set_group_name,
+            #  test_group_name_available_in_inference_pass,
+            #  test_groupby_multi_timezone
             object.__setattr__(group, "name", key)
 
             # group might be modified
diff --git a/pandas/tests/groupby/test_groupby.py b/pandas/tests/groupby/test_groupby.py
index e225ff5a0fa43..97e88a8545aa5 100644
--- a/pandas/tests/groupby/test_groupby.py
+++ b/pandas/tests/groupby/test_groupby.py
@@ -65,7 +65,7 @@ def test_groupby_std_datetimelike():
 
 
 @pytest.mark.parametrize("dtype", ["int64", "int32", "float64", "float32"])
-def test_basic(dtype):
+def test_basic_aggregations(dtype):
     data = Series(np.arange(9) // 3, index=np.arange(9), dtype=dtype)
 
     index = np.arange(9)
@@ -102,7 +102,13 @@ def test_basic(dtype):
         grouped.aggregate({"one": np.mean, "two": np.std})
 
     group_constants = {0: 10, 1: 20, 2: 30}
-    agged = grouped.agg(lambda x: group_constants[x.name] + x.mean())
+    msg = (
+        "Pinning the groupby key to each group in SeriesGroupBy.agg is deprecated, "
+        "and cases that relied on it will raise in a future version"
+    )
+    with tm.assert_produces_warning(FutureWarning, match=msg):
+        # GH#41090
+        agged = grouped.agg(lambda x: group_constants[x.name] + x.mean())
     assert agged[1] == 21
 
     # corner cases