BUG: Break reference from grouping level to MI

pandas/core/indexes/multi.py

@@ -1256,6 +1256,10 @@ def _get_grouper_for_level(self, mapper, level):
         if len(uniques) < len(level_index):
             # Remove unobserved levels from level_index
             level_index = level_index.take(uniques)
+        else:
+            # break references back to us so that setting the name
+            # on the output of a groupby doesn't reflect back here.
+            level_index = level_index.copy()
 
         if len(level_index):
             grouper = level_index.take(codes)

pandas/tests/groupby/test_apply.py

@@ -775,3 +775,20 @@ def most_common_values(df):
         ["17661101"], index=pd.DatetimeIndex(["2015-02-24"], name="day"), name="userId"
     )
     tm.assert_series_equal(result, expected)
+
+
+@pytest.mark.parametrize("category", [False, True])
+def test_apply_multi_level_name(category):
+    # https://github.com/pandas-dev/pandas/issues/31068
+    b = [1, 2] * 5
+    if category:
+        b = pd.Categorical(b, categories=[1, 2, 3])
+    df = pd.DataFrame(
+        {"A": np.arange(10), "B": b, "C": list(range(10)), "D": list(range(10))}
+    ).set_index(["A", "B"])
+    result = df.groupby("B").apply(lambda x: x.sum())
+    expected = pd.DataFrame(
+        {"C": [20, 25], "D": [20, 25]}, index=pd.Index([1, 2], name="B")
+    )
+    tm.assert_frame_equal(result, expected)
+    assert df.index.names == ["A", "B"]