diff --git a/doc/source/whatsnew/v1.2.0.rst b/doc/source/whatsnew/v1.2.0.rst
index b16ca0a80c5b4..74c683638c654 100644
--- a/doc/source/whatsnew/v1.2.0.rst
+++ b/doc/source/whatsnew/v1.2.0.rst
@@ -132,6 +132,7 @@ Plotting
 Groupby/resample/rolling
 ^^^^^^^^^^^^^^^^^^^^^^^^
 
+- Bug in :meth:`DataFrameGroupBy.apply` that would some times throw an erroneous ``ValueError`` if the grouping axis had duplicate entries (:issue:`16646`)
 -
 -
 
diff --git a/pandas/core/groupby/ops.py b/pandas/core/groupby/ops.py
index 3aaeef3b63760..2cb4674b2e33a 100644
--- a/pandas/core/groupby/ops.py
+++ b/pandas/core/groupby/ops.py
@@ -211,7 +211,7 @@ def apply(self, f: F, data: FrameOrSeries, axis: int = 0):
             # group might be modified
             group_axes = group.axes
             res = f(group)
-            if not _is_indexed_like(res, group_axes):
+            if not _is_indexed_like(res, group_axes, axis):
                 mutated = True
             result_values.append(res)
 
@@ -897,13 +897,13 @@ def agg_series(
         return grouper.get_result()
 
 
-def _is_indexed_like(obj, axes) -> bool:
+def _is_indexed_like(obj, axes, axis: int) -> bool:
     if isinstance(obj, Series):
         if len(axes) > 1:
             return False
-        return obj.index.equals(axes[0])
+        return obj.axes[axis].equals(axes[axis])
     elif isinstance(obj, DataFrame):
-        return obj.index.equals(axes[0])
+        return obj.axes[axis].equals(axes[axis])
 
     return False
 
diff --git a/pandas/tests/groupby/test_apply.py b/pandas/tests/groupby/test_apply.py
index 525a6fe2637c3..665cd12225ad7 100644
--- a/pandas/tests/groupby/test_apply.py
+++ b/pandas/tests/groupby/test_apply.py
@@ -63,15 +63,8 @@ def test_apply_trivial():
     tm.assert_frame_equal(result, expected)
 
 
-@pytest.mark.xfail(
-    reason="GH#20066; function passed into apply "
-    "returns a DataFrame with the same index "
-    "as the one to create GroupBy object."
-)
 def test_apply_trivial_fail():
     # GH 20066
-    # trivial apply fails if the constant dataframe has the same index
-    # with the one used to create GroupBy object.
     df = pd.DataFrame(
         {"key": ["a", "a", "b", "b", "a"], "data": [1.0, 2.0, 3.0, 4.0, 5.0]},
         columns=["key", "data"],
@@ -1044,3 +1037,23 @@ def test_apply_with_date_in_multiindex_does_not_convert_to_timestamp():
     tm.assert_frame_equal(result, expected)
     for val in result.index.levels[1]:
         assert type(val) is date
+
+
+def test_apply_by_cols_equals_apply_by_rows_transposed():
+    # GH 16646
+    # Operating on the columns, or transposing and operating on the rows
+    # should give the same result. There was previously a bug where the
+    # by_rows operation would work fine, but by_cols would throw a ValueError
+
+    df = pd.DataFrame(
+        np.random.random([6, 4]),
+        columns=pd.MultiIndex.from_product([["A", "B"], [1, 2]]),
+    )
+
+    by_rows = df.T.groupby(axis=0, level=0).apply(
+        lambda x: x.droplevel(axis=0, level=0)
+    )
+    by_cols = df.groupby(axis=1, level=0).apply(lambda x: x.droplevel(axis=1, level=0))
+
+    tm.assert_frame_equal(by_cols, by_rows.T)
+    tm.assert_frame_equal(by_cols, df)
diff --git a/pandas/tests/groupby/test_function.py b/pandas/tests/groupby/test_function.py
index cbfba16223f74..42945be923fa0 100644
--- a/pandas/tests/groupby/test_function.py
+++ b/pandas/tests/groupby/test_function.py
@@ -940,10 +940,6 @@ def test_frame_describe_multikey(tsframe):
     groupedT = tsframe.groupby({"A": 0, "B": 0, "C": 1, "D": 1}, axis=1)
     result = groupedT.describe()
     expected = tsframe.describe().T
-    expected.index = pd.MultiIndex(
-        levels=[[0, 1], expected.index],
-        codes=[[0, 0, 1, 1], range(len(expected.index))],
-    )
     tm.assert_frame_equal(result, expected)