BUG: Break reference from grouping level to MI

Closes pandas-dev#31068

Author: TomAugspurger

pandas/core/indexes/multi.py

@@ -1235,6 +1235,9 @@ def _set_names(self, names, level=None, validate=True):
     def _get_grouper_for_level(self, mapper, level):
         indexer = self.codes[level]
         level_index = self.levels[level]
+        # break references back to ourself so that setting the name
+        # on the output of a groupby doesn't reflect back here.
+        level_index = level_index.copy()
 
         if mapper is not None:
             # Handle group mapping function and return

pandas/tests/groupby/test_apply.py

@@ -752,3 +752,21 @@ def most_common_values(df):
         ["17661101"], index=pd.DatetimeIndex(["2015-02-24"], name="day"), name="userId"
     )
     tm.assert_series_equal(result, expected)
+
+
+def test_apply_multi_level_name():
+    # https://github.com/pandas-dev/pandas/issues/31068
+    df = pd.DataFrame(
+        {
+            "A": np.arange(10),
+            "B": [1, 2] * 5,
+            "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"]