diff --git a/doc/source/whatsnew/v1.2.0.rst b/doc/source/whatsnew/v1.2.0.rst
index e65daa439a225..7ff946574c778 100644
--- a/doc/source/whatsnew/v1.2.0.rst
+++ b/doc/source/whatsnew/v1.2.0.rst
@@ -311,6 +311,7 @@ Groupby/resample/rolling
 - Bug in :meth:`DataFrameGroupby.apply` would drop a :class:`CategoricalIndex` when grouped on. (:issue:`35792`)
 - Bug when subsetting columns on a :class:`~pandas.core.groupby.DataFrameGroupBy` (e.g. ``df.groupby('a')[['b']])``) would reset the attributes ``axis``, ``dropna``, ``group_keys``, ``level``, ``mutated``, ``sort``, and ``squeeze`` to their default values. (:issue:`9959`)
 - Bug in :meth:`DataFrameGroupby.tshift` failing to raise ``ValueError`` when a frequency cannot be inferred for the index of a group (:issue:`35937`)
+- 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 537feace59fcb..a58f28880945f 100644
--- a/pandas/core/groupby/generic.py
+++ b/pandas/core/groupby/generic.py
@@ -1694,6 +1694,7 @@ def _wrap_transformed_output(
         """
         indexed_output = {key.position: val for key, val in output.items()}
         columns = Index(key.label for key in output)
+        columns.name = self.obj.columns.name
 
         result = self.obj._constructor(indexed_output)
         result.columns = columns
diff --git a/pandas/tests/groupby/test_groupby.py b/pandas/tests/groupby/test_groupby.py
index eec9e8064d584..e0196df7ceac0 100644
--- a/pandas/tests/groupby/test_groupby.py
+++ b/pandas/tests/groupby/test_groupby.py
@@ -2111,3 +2111,26 @@ def test_subsetting_columns_keeps_attrs(klass, attr, value):
     expected = df.groupby("a", **{attr: value})
     result = expected[["b"]] if klass is DataFrame else expected["b"]
     assert getattr(result, attr) == getattr(expected, attr)
+
+
+@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)