diff --git a/pandas/tests/groupby/test_apply.py b/pandas/tests/groupby/test_apply.py
index ae73ddc001dc1..62d4a0ddcc0f5 100644
--- a/pandas/tests/groupby/test_apply.py
+++ b/pandas/tests/groupby/test_apply.py
@@ -227,6 +227,22 @@ def fast(group):
     tm.assert_frame_equal(fast_df, slow_df)
 
 
+def test_apply_fast_slow_identical_index():
+    # GH#44803
+    df = DataFrame(
+        {
+            "name": ["Alice", "Bob", "Carl"],
+            "age": [20, 21, 20],
+        }
+    ).set_index("name")
+
+    grp_by_same_value = df.groupby(["age"], group_keys=False).apply(lambda group: group)
+    grp_by_copy = df.groupby(["age"], group_keys=False).apply(
+        lambda group: group.copy()
+    )
+    tm.assert_frame_equal(grp_by_same_value, grp_by_copy)
+
+
 @pytest.mark.parametrize(
     "func",
     [
@@ -1463,3 +1479,37 @@ def f_4(grp):
     e.loc["Pony"] = np.nan
     e.name = None
     tm.assert_series_equal(result, e)
+
+
+def test_nonreducer_nonstransform():
+    # GH3380, GH60619
+    # Was originally testing mutating in a UDF; now kept as an example
+    # of using apply with a nonreducer and nontransformer.
+    df = DataFrame(
+        {
+            "cat1": ["a"] * 8 + ["b"] * 6,
+            "cat2": ["c"] * 2
+            + ["d"] * 2
+            + ["e"] * 2
+            + ["f"] * 2
+            + ["c"] * 2
+            + ["d"] * 2
+            + ["e"] * 2,
+            "val": np.random.default_rng(2).integers(100, size=14),
+        }
+    )
+
+    def f(x):
+        x = x.copy()
+        x["rank"] = x.val.rank(method="min")
+        return x.groupby("cat2")["rank"].min()
+
+    expected = DataFrame(
+        {
+            "cat1": list("aaaabbb"),
+            "cat2": list("cdefcde"),
+            "rank": [3.0, 2.0, 5.0, 1.0, 2.0, 4.0, 1.0],
+        }
+    ).set_index(["cat1", "cat2"])["rank"]
+    result = df.groupby("cat1").apply(f)
+    tm.assert_series_equal(result, expected)
diff --git a/pandas/tests/groupby/test_apply_mutate.py b/pandas/tests/groupby/test_apply_mutate.py
deleted file mode 100644
index ee0912175f024..0000000000000
--- a/pandas/tests/groupby/test_apply_mutate.py
+++ /dev/null
@@ -1,65 +0,0 @@
-import numpy as np
-
-import pandas as pd
-import pandas._testing as tm
-
-
-def test_group_by_copy():
-    # GH#44803
-    df = pd.DataFrame(
-        {
-            "name": ["Alice", "Bob", "Carl"],
-            "age": [20, 21, 20],
-        }
-    ).set_index("name")
-
-    grp_by_same_value = df.groupby(["age"], group_keys=False).apply(lambda group: group)
-    grp_by_copy = df.groupby(["age"], group_keys=False).apply(
-        lambda group: group.copy()
-    )
-    tm.assert_frame_equal(grp_by_same_value, grp_by_copy)
-
-
-def test_mutate_groups():
-    # GH3380
-
-    df = pd.DataFrame(
-        {
-            "cat1": ["a"] * 8 + ["b"] * 6,
-            "cat2": ["c"] * 2
-            + ["d"] * 2
-            + ["e"] * 2
-            + ["f"] * 2
-            + ["c"] * 2
-            + ["d"] * 2
-            + ["e"] * 2,
-            "cat3": [f"g{x}" for x in range(1, 15)],
-            "val": np.random.default_rng(2).integers(100, size=14),
-        }
-    )
-
-    def f(x):
-        x = x.copy()
-        x["rank"] = x.val.rank(method="min")
-        return x.groupby("cat2")["rank"].min()
-
-    expected = pd.DataFrame(
-        {
-            "cat1": list("aaaabbb"),
-            "cat2": list("cdefcde"),
-            "rank": [3.0, 2.0, 5.0, 1.0, 2.0, 4.0, 1.0],
-        }
-    ).set_index(["cat1", "cat2"])["rank"]
-    result = df.groupby("cat1").apply(f)
-    tm.assert_series_equal(result, expected)
-
-
-def test_no_mutate_but_looks_like():
-    # GH 8467
-    # first show's mutation indicator
-    # second does not, but should yield the same results
-    df = pd.DataFrame({"key": [1, 1, 1, 2, 2, 2, 3, 3, 3], "value": range(9)})
-
-    result1 = df.groupby("key", group_keys=True).apply(lambda x: x[:].value)
-    result2 = df.groupby("key", group_keys=True).apply(lambda x: x.value)
-    tm.assert_series_equal(result1, result2)