Hi I would like to give permission to just certain groups of players for certain weapons. How should I do that?

I wanted to set a balance for certain weapon for all group or "Group Default" but it seems that it is meant for a single player.

You should be able to grant the permission that actually enables the scaling to just a specific group, if that's what you're looking to do. However, what you won't be able to do, is have different scaling options for *multiple* groups. It wasn't really designed to be permission based.