How to add a custom taxonomy to show up in a custom post type menu?

So to answear my own question after much googeling:

In the args array I commented out the option for show_in_menu, so that custom post type creates the menu itself. Then I didn’t add the admin menu, but just the submenu (the hook still remains).

    add_action('admin_menu','bsp_plugin_menu');

And the first parameter is the one that shows it in the menu and it is a name of the custom post type:

    add_submenu_page(**'edit.php?post_type=badge'**, 'Add new students', 'Add new students', 'manage_options','add-new-students','bsp_students_add');

And now it shows in the menu with all the taxonomies and custom post.