Allow logged in user to change his own user group using front end form

Question

I'd like to allow a user to change his user group via front end form. I tried passing the groupId as the destination group, but it doesn't seem to work.

This is what I was passing:

<input type="hidden" name="action" value="users/save-user">
<input type="hidden" name="userId" value="{{ currentUser.id }}">
<input type="hidden" name="groupId" value="2">  

I came across this post and am thinking that it's not possible to do without a plugin.