as far as i know, variables you assign to groups and units as their identifiers cannot be returned through scripts as discrete names.
put another way, think of the group as a colour - the name you give it is "red", but asking flashpoint what colour the group is will not return 'red', it can only show you the actual colour.
does that make sense?
to develop on your question, to find out the group to which a unit activating a trigger belongs, in the on-activation line put
mygroup = group (thislist select 0)
you can now use the global variable mygroup to point to whichever group that activating unit is in.