The Backup Alt-F service relies on all users belonging to the backup group, so it will probably stop working as expected if that doesn't happens. That's why the 'backup' group doesn't appears in the Setup Users webUI -- its a system group!
But if nonetheless you which to do it you have to resort to the command line:
[root@dns-320l]# id mlima # shows which groups 'mlima' belongs to
uid=1001(mlima) gid=100(users) groups=100(users),34(backup)
[root@dns-320l]# delgroup mlima backup # delete mlima from backup group
[root@dns-320l]# id mlima
uid=1001(mlima) gid=100(users) groups=100(users)
[root@dns-320l]# addgroup mlima backup # re-add mlima to backup group
So you want to have a John alias called Timmy, is that it? Whenever John or Timmy login both will have the same home folder.
That can be done but is highly non-standard. The main reason is related with file permissions and ownership. If Tommy logs in and create file foo, that file will belong to him; if latter John logs in and tries to edit that file that will fail, as it doesn't owns the file. That is a simple situation -- depending on how the home folder permissions are set, one of the alias user will not even be able to login.
What you can to do is to create another folder, say Shared, belonging to the group JohnTimmy with group read/write/browse permissions, and make John and Timmy belong to that group. Then, any one of them will be able to access the Shared folder.
Read the Permissions online page (accessible through the Folders Browse webUI) and read a linux tutorial on file permissions and ownership.
If however that is not what you want, but you just want that Jimmy's home folder to be named Users/John (without a John users existing) (can't imagine why, but the need is yours), you have to edit the /etc/password file and change the corresponding entry. You might have a lot of issues, as some programs assume that a users home folder is called after its nickname.
Alt-F is non standard in that respect, it uses the user full name instead of the user nick name to name its home folder. So, 'jcard' (Joao Cardoso) home folder is not called /home/jcard but "/home/Joao Cardoso". Occasionally some programs choke with that non-conformity to the traditional unix standard -- but there nothing in the standard that prohibits it, it's just a 45 years long convention that I decided to break -- it was about time!