Please try now.
I'm not sure if this is exactly what OP needs though. It adds a permission for a user to create a group and assign it to a category. Not exactly the same thing as approval.
I ask a lot of questions - heh. Sorry.
I'm not certain if this is something that can be done as plugin or if this is something that would have to to be done as a core enhancement - so I'm asking here before I go to far down the rabbit hole.
I'd like to do two things for groups. The first, add the option of an external bitmap image. The second is add the option of adding to or replacing the Font Awesome options with Font Custom ( http://fontcustom.com/ ) which is essentially roll your own Font Awesome.
My rationale is that I'm going to abuse groups slightly to provide two requirements that have been imposed on me ( Badges and Forum Titles ) by using Groups for both. Those with Images , badges, those without, titles.
Anyhow. Is this an enhancement to the codebase or something I can do with a plugin? Pointers would be great.