diff --git a/include/pybind11/cast.h b/include/pybind11/cast.h
index 9da63772e..af6705b62 100644
--- a/include/pybind11/cast.h
+++ b/include/pybind11/cast.h
@@ -888,6 +888,10 @@ struct handle_type_name<dict> {
     static constexpr auto name = const_name("dict");
 };
 template <>
+struct handle_type_name<anyset> {
+    static constexpr auto name = const_name("set");
+};
+template <>
 struct handle_type_name<set> {
     static constexpr auto name = const_name("set");
 };
@@ -896,10 +900,6 @@ struct handle_type_name<frozenset> {
     static constexpr auto name = const_name("frozenset");
 };
 template <>
-struct handle_type_name<anyset> {
-    static constexpr auto name = const_name("anyset");
-};
-template <>
 struct handle_type_name<str> {
     static constexpr auto name = const_name("str");
 };
diff --git a/tests/test_pytypes.py b/tests/test_pytypes.py
index 2b2027316..2824d29f9 100644
--- a/tests/test_pytypes.py
+++ b/tests/test_pytypes.py
@@ -121,7 +121,7 @@ def test_set(capture, doc):
     assert m.anyset_contains({"foo"}, "foo")
 
     assert doc(m.get_set) == "get_set() -> set"
-    assert doc(m.print_anyset) == "print_anyset(arg0: anyset) -> None"
+    assert doc(m.print_anyset) == "print_anyset(arg0: set) -> None"
 
 
 def test_frozenset(capture, doc):