Assuming you're using the core `users/save-user` action to edit users (as per the official [User Profile form example](https://craftcms.com/knowledge-base/front-end-user-accounts#user-profile-form)) you can add a form input named `groups` to save group assignments from the frontend form.

_Note that this only works when editing an existing user's profile, i.e. not when registering new users._

Before saving the user, Craft will handle permission checking for you (again, assuming you're posting the form to the core `users/save-user` action). Still, it's probably a good idea to make sure that only groups that the current user is able to save assignments for, are exposed as options in the form. Here's one way to achieve that – first pulling all user groups via the `craft.app.userGroups` service, and then filtering those groups, using the [`currentUser.can()`][1] method to check that the current user has the necessary `assignUserGroup:[UserGroupUID]` permission for each group:

```twig
{% set userGroups = craft.app.userGroups.allGroups()|filter(group => currentUser.can("assignUserGroup:#{group.uid}")) %}
```

Once you have your `userGroups` array, you can build a dropdown:  

```twig
<form method="post" accept-charset="UTF-8" enctype="multipart/form-data">
    ...
    <label for="group">User group</label>
    <select id="group" name="groups[]">
        {% for userGroup in userGroups %}
            <option value="{{ userGroup.id }}"{% if user.isInGroup(userGroup) %} selected{% endif %}>{{ userGroup.name }}</option>
        {% endfor %}
    </select>
    ...
</form>
```

Or if you prefer, checkboxes. Notice the hidden input `name="groups"` above the actual checkbox inputs – that's there to make it possible to submit the form without any groups checked:  

```twig
<input type="hidden" name="groups" value/>
<fieldset>
    <legend>User groups</legend>
    {% for userGroup in userGroups %}
        <label>
            <input type="checkbox" name="groups[]" value="{{ userGroup.id }}"{% if user.isInGroup(userGroup) %} checked{% endif %} />
            <span>{{ userGroup.name }}</span>
        </label>
    {% endfor %}
</fieldset>
```

Or radio buttons:

```twig
<fieldset>
    <legend>User group</legend>
    {% for userGroup in userGroups %}
        <label>
            <input type="radio" name="groups[]" value="{{ userGroup.id }}"{% if user.isInGroup(userGroup) %} checked{% endif %} />
            <span>{{ userGroup.name }}</span>
        </label>
    {% endfor %}
</fieldset>
```


  [1]: https://craftcms.com/docs/3.x/user-management.html#checking-permissions