Duane wrote: "I have no problem using the GIO (envelope) icon on all tabs."
It might be a good default. And since each tab is also a tooltip, you can "hover" the pointer over the envelopes to see which is which (unless you're using a touchscreen device).
Maybe one of those things where group owners are the best judges of what their groups should do.
I would venture to say that many of the users are members of only one group. Mark probably has that statistic or could generate it in a few keystrokes. For those users, a group-specific icon makes sense.
For people in GMF and Beta and the other spots where function and administration are discussed there are arguments on both sides.