It's possible but I think you have to write php code to do it. You could create a scheduled task script do it, but I think you'd need some field in the database to keep track of whether or not the PM had been sent for each user (maybe a custom profile field would work).
I guess you're probably using a promotion to add the group? Then maybe another way would be to hack the promotion code to send the message when it happens. Edit: Actually someone has done it for vb3.6:
https://vborg.vbsupport.ru/showthread.php?t=140629 and if you look near the end of the thread, someone has posted a change to allow it to work with 3.8, so you might try it with a 4.X version (assuming the "PM on promotion" option applies to your situation).