How to show pop-up menu when an IconButton is clicked Flutter

I am trying to show a pop-up menu appear when an IconButton() is clicked I added a PopupMenuButton() to the on pressed method of the IconButton as you can see below:

but it does not show any pop-up menu when clicked.
If there’s anything I’m doing wrong I hope some one can point it out for me.
Here is the code:

import 'package:flutter/foundation.dart';
import 'package:flutter/material.dart';

String _selectedMenu = '';

enum Menu { itemOne, itemTwo, itemThree, itemFour }

class ListTileWithIcon extends StatefulWidget {
  const ListTileWithIcon({super.key});

  @override
  State<ListTileWithIcon> createState() => _ListTileWithIconState();
}

class _ListTileWithIconState extends State<ListTileWithIcon> {
  @override
  Widget build(BuildContext context) {
    return ListTile(
      leading: const Icon(Icons.person),
      title: const Text("The ListTile"),
      trailing: IconButton(
        // key: _icnBtnKey,
        splashRadius: 20,
        icon: const Icon(
          Icons.more_vert_rounded,
          size: 20.0,
          color: Color.fromARGB(255, 135, 0, 212),
        ),
        onPressed: () {
          // showPopUpMenuAtTap();
          if (kDebugMode) {
            print("Inside icon button's clickt");
          }

          PopupMenuButton<Menu>(
            // Callback that sets the selected popup menu item.
            onSelected: (Menu item) {
              setState(() {
                _selectedMenu = item.name;
              });
            },
            itemBuilder: (BuildContext context) => <PopupMenuEntry<Menu>>[
              const PopupMenuItem<Menu>(
                value: Menu.itemOne,
                child: Text('Item 1'),
              ),
              const PopupMenuItem<Menu>(
                value: Menu.itemTwo,
                child: Text('Item 2'),
              ),
              const PopupMenuItem<Menu>(
                value: Menu.itemThree,
                child: Text('Item 3'),
              ),
              const PopupMenuItem<Menu>(
                value: Menu.itemFour,
                child: Text('Item 4'),
              ),
            ],
          );
        },
      ),
      onTap: () {},
    );
  }
}

Thanks in advance.