I have a group of "Past Members" (ones that no longer meet the age restrictions of our forums). They are set to "Can access site?" = No. 

Is it possible to display a message specifically to this group after a login attempt has been made? 

We would like to explain why they can't login, and invite them to join the organization (now that they are old enough). 

If it is possible, how do I do this? 

 

Thank you. 

Theme template core->front->global->error.

Insert this code in <div id='elErrorMessage' class='ipsPos_center'> of wherever you see fit:

{{if \IPS\Member::loggedIn()->inGroup(X)}}
Your message
{{endif}}

Replace X with the ID of the "Past Members" group.

There is also that theme tip on how to make custom error pages. But you need Pages.
Edit: Although to display specifically to that group you'll need the code -FP put above. You could do a manual HTML error page or as he suggested.