diff --git a/integration/client/snapshot_test.go b/integration/client/snapshot_test.go
index d5eed0a5e..1aca97147 100644
--- a/integration/client/snapshot_test.go
+++ b/integration/client/snapshot_test.go
@@ -47,5 +47,5 @@ func TestSnapshotterClient(t *testing.T) {
 	if runtime.GOOS == "windows" {
 		t.Skip("snapshots not yet supported on Windows")
 	}
-	testsuite.SnapshotterSuite(t, "SnapshotterClient", newSnapshotter)
+	testsuite.SnapshotterSuite(t, DefaultSnapshotter, newSnapshotter)
 }
diff --git a/snapshots/overlay/overlay_test.go b/snapshots/overlay/overlay_test.go
index 279613935..13493c45e 100644
--- a/snapshots/overlay/overlay_test.go
+++ b/snapshots/overlay/overlay_test.go
@@ -57,7 +57,7 @@ func TestOverlay(t *testing.T) {
 	for optsName, opts := range optTestCases {
 		t.Run(optsName, func(t *testing.T) {
 			newSnapshotter := newSnapshotterWithOpts(opts...)
-			testsuite.SnapshotterSuite(t, "Overlay", newSnapshotter)
+			testsuite.SnapshotterSuite(t, "overlayfs", newSnapshotter)
 			t.Run("TestOverlayMounts", func(t *testing.T) {
 				testOverlayMounts(t, newSnapshotter)
 			})
diff --git a/snapshots/testsuite/issues.go b/snapshots/testsuite/issues.go
index 6a5fe49a0..eea87767e 100644
--- a/snapshots/testsuite/issues.go
+++ b/snapshots/testsuite/issues.go
@@ -114,23 +114,33 @@ func checkChown(ctx context.Context, t *testing.T, sn snapshots.Snapshotter, wor
 
 // checkRename
 // https://github.com/docker/docker/issues/25409
-func checkRename(ctx context.Context, t *testing.T, sn snapshots.Snapshotter, work string) {
-	t.Skip("rename test still fails on some kernels with overlay")
-	l1Init := fstest.Apply(
-		fstest.CreateDir("/dir1", 0700),
-		fstest.CreateDir("/somefiles", 0700),
-		fstest.CreateFile("/somefiles/f1", []byte("was here first!"), 0644),
-		fstest.CreateFile("/somefiles/f2", []byte("nothing interesting"), 0644),
-	)
-	l2Init := fstest.Apply(
-		fstest.Rename("/dir1", "/dir2"),
-		fstest.CreateFile("/somefiles/f1-overwrite", []byte("new content 1"), 0644),
-		fstest.Rename("/somefiles/f1-overwrite", "/somefiles/f1"),
-		fstest.Rename("/somefiles/f2", "/somefiles/f3"),
-	)
+func checkRename(ss string) func(ctx context.Context, t *testing.T, sn snapshots.Snapshotter, work string) {
+	return func(ctx context.Context, t *testing.T, sn snapshots.Snapshotter, work string) {
+		l1Init := fstest.Apply(
+			fstest.CreateDir("/dir1", 0700),
+			fstest.CreateDir("/somefiles", 0700),
+			fstest.CreateFile("/somefiles/f1", []byte("was here first!"), 0644),
+			fstest.CreateFile("/somefiles/f2", []byte("nothing interesting"), 0644),
+		)
 
-	if err := checkSnapshots(ctx, sn, work, l1Init, l2Init); err != nil {
-		t.Fatalf("Check snapshots failed: %+v", err)
+		var applier []fstest.Applier
+		if ss != "overlayfs" {
+			// With neither OVERLAY_FS_REDIRECT_DIR nor redirect_dir,
+			// renaming the directory on the lower directory doesn't work on overlayfs.
+			// https://github.com/torvalds/linux/blob/v5.18/Documentation/filesystems/overlayfs.rst#renaming-directories
+			applier = append(applier, fstest.Rename("/dir1", "/dir2"))
+		}
+		applier = append(
+			applier,
+			fstest.CreateFile("/somefiles/f1-overwrite", []byte("new content 1"), 0644),
+			fstest.Rename("/somefiles/f1-overwrite", "/somefiles/f1"),
+			fstest.Rename("/somefiles/f2", "/somefiles/f3"),
+		)
+		l2Init := fstest.Apply(applier...)
+
+		if err := checkSnapshots(ctx, sn, work, l1Init, l2Init); err != nil {
+			t.Fatalf("Check snapshots failed: %+v", err)
+		}
 	}
 }
 
diff --git a/snapshots/testsuite/testsuite.go b/snapshots/testsuite/testsuite.go
index d62c41496..cf793d1cd 100644
--- a/snapshots/testsuite/testsuite.go
+++ b/snapshots/testsuite/testsuite.go
@@ -62,7 +62,6 @@ func SnapshotterSuite(t *testing.T, name string, snapshotterFn SnapshotterFunc)
 	t.Run("RemoveIntermediateSnapshot", makeTest(name, snapshotterFn, checkRemoveIntermediateSnapshot))
 	t.Run("DeletedFilesInChildSnapshot", makeTest(name, snapshotterFn, checkDeletedFilesInChildSnapshot))
 	t.Run("MoveFileFromLowerLayer", makeTest(name, snapshotterFn, checkFileFromLowerLayer))
-	t.Run("Rename", makeTest(name, snapshotterFn, checkRename))
 
 	t.Run("ViewReadonly", makeTest(name, snapshotterFn, checkSnapshotterViewReadonly))
 
@@ -70,10 +69,16 @@ func SnapshotterSuite(t *testing.T, name string, snapshotterFn SnapshotterFunc)
 	t.Run("CloseTwice", makeTest(name, snapshotterFn, closeTwice))
 	t.Run("RootPermission", makeTest(name, snapshotterFn, checkRootPermission))
 
+	// Different snapshotters behave slightly differently in the tests below.
+	t.Run("Rename", makeTest(name, snapshotterFn, checkRename(name)))
 	t.Run("128LayersMount", makeTest(name, snapshotterFn, check128LayersMount(name)))
 }
 
-func makeTest(name string, snapshotterFn func(ctx context.Context, root string) (snapshots.Snapshotter, func() error, error), fn func(ctx context.Context, t *testing.T, snapshotter snapshots.Snapshotter, work string)) func(t *testing.T) {
+func makeTest(
+	snapshotter string,
+	snapshotterFn func(ctx context.Context, root string) (snapshots.Snapshotter, func() error, error),
+	fn func(ctx context.Context, t *testing.T, snapshotter snapshots.Snapshotter, work string),
+) func(t *testing.T) {
 	return func(t *testing.T) {
 		t.Parallel()
 
@@ -85,7 +90,7 @@ func makeTest(name string, snapshotterFn func(ctx context.Context, root string)
 		// 		work/ -> passed to test functions
 		// 		root/ -> passed to snapshotter
 		//
-		tmpDir, err := os.MkdirTemp("", "snapshot-suite-"+name+"-")
+		tmpDir, err := os.MkdirTemp("", "snapshot-suite-"+snapshotter+"-")
 		if err != nil {
 			t.Fatal(err)
 		}