I am not "allowed" to add users directly to the administrators or remote desktop user groups due to internal reasons (This is all local).
I have successfully created the new local group and added my users to that. I then successfully added my new local group to the administrator and remote desktop users group. (Again, still local).
Even though I have created the new local group, added the users to that new group, then added that new local group the administrator and remote desktop users group, I cannot seem to do things like RDP. (I personally am not that great when it comes to local group inheritance)