How to add new user to group during registration

I am somewhat a newbie in Drupal but I have done allot. What I am attempting is to add a new user to a Group (not OG) during registration.

I have tried many rules and flagging but fail to get a result. There is allot of documentation for adding members in OG but I can find anything for just Groups.

I have 8 different Groups set up, all working as intended. I have a separate registration page for each Group. 

Users are in a blocked status till leader of the group unblocks them. (this part all works fine)

I would like to have them added to the Group during the registration with only member access till the leader of each Group changes that new users Group Role.

Would also like to have it so no group member can join another Group without being removed from the one he/she belongs to.