diff --git a/doc/source/whatsnew/v1.2.0.rst b/doc/source/whatsnew/v1.2.0.rst
index b16ca0a80c5b4..010c8a22dcc9f 100644
--- a/doc/source/whatsnew/v1.2.0.rst
+++ b/doc/source/whatsnew/v1.2.0.rst
@@ -132,9 +132,8 @@ Plotting
 Groupby/resample/rolling
 ^^^^^^^^^^^^^^^^^^^^^^^^
 
+- Bug in :meth:`DataFrame.groupby` does not always maintain column index name for ``any``, ``all``, ``bfill``, ``ffill``, ``shift`` (:issue:`29764`)
 -
--
-
 
 Reshaping
 ^^^^^^^^^
diff --git a/pandas/core/groupby/generic.py b/pandas/core/groupby/generic.py
index c50b753cf3293..bc46cc041cbf3 100644
--- a/pandas/core/groupby/generic.py
+++ b/pandas/core/groupby/generic.py
@@ -1728,9 +1728,10 @@ def _wrap_aggregated_output(
         -------
         DataFrame
         """
+        idx_name = output.pop("idx_name", None)
         indexed_output = {key.position: val for key, val in output.items()}
         name = self._obj_with_exclusions._get_axis(1 - self.axis).name
-        columns = Index([key.label for key in output], name=name)
+        columns = Index([key.label for key in output], name=idx_name)
 
         result = self.obj._constructor(indexed_output)
         result.columns = columns
@@ -1762,8 +1763,9 @@ def _wrap_transformed_output(
         -------
         DataFrame
         """
+        idx_name = output.pop("idx_name", None)
         indexed_output = {key.position: val for key, val in output.items()}
-        columns = Index(key.label for key in output)
+        columns = Index([key.label for key in output], name=idx_name)
 
         result = self.obj._constructor(indexed_output)
         result.columns = columns
diff --git a/pandas/core/groupby/groupby.py b/pandas/core/groupby/groupby.py
index ac45222625569..5a20af9607cd9 100644
--- a/pandas/core/groupby/groupby.py
+++ b/pandas/core/groupby/groupby.py
@@ -2450,8 +2450,11 @@ def _get_cythonized_result(
         grouper = self.grouper
 
         labels, _, ngroups = grouper.group_info
-        output: Dict[base.OutputKey, np.ndarray] = {}
+        output: Dict[Union[base.OutputKey, str], Union[np.ndarray, str]] = {}
         base_func = getattr(libgroupby, how)
+        obj = self._selected_obj
+        if isinstance(obj, DataFrame):
+            output["idx_name"] = obj.columns.name
 
         error_msg = ""
         for idx, obj in enumerate(self._iterate_slices()):
diff --git a/pandas/tests/groupby/test_groupby.py b/pandas/tests/groupby/test_groupby.py
index 8c51ebf89f5c0..19dd8a1cee355 100644
--- a/pandas/tests/groupby/test_groupby.py
+++ b/pandas/tests/groupby/test_groupby.py
@@ -2069,3 +2069,26 @@ def test_group_on_two_row_multiindex_returns_one_tuple_key():
     assert len(result) == 1
     key = (1, 2)
     assert (result[key] == expected[key]).all()
+
+
+@pytest.mark.parametrize("func", ["sum", "any", "shift"])
+def test_groupby_column_index_name_lost(func):
+    # GH: 29764 groupby loses index sometimes
+    expected = pd.Index(["a"], name="idx")
+    df = pd.DataFrame([[1]], columns=expected)
+    df_grouped = df.groupby([1])
+    result = getattr(df_grouped, func)().columns
+    tm.assert_index_equal(result, expected)
+
+
+@pytest.mark.parametrize("func", ["ffill", "bfill"])
+def test_groupby_column_index_name_lost_fill_funcs(func):
+    # GH: 29764 groupby loses index sometimes
+    df = pd.DataFrame(
+        [[1, 1.0, -1.0], [1, np.nan, np.nan], [1, 2.0, -2.0]],
+        columns=pd.Index(["type", "a", "b"], name="idx"),
+    )
+    df_grouped = df.groupby(["type"])[["a", "b"]]
+    result = getattr(df_grouped, func)().columns
+    expected = pd.Index(["a", "b"], name="idx")
+    tm.assert_index_equal(result, expected)