Ha ha Yes this is absurd fun. The word 'group' is singular and 'groups' is the plural. All y'alls meaning more than one group, is plural.Mmmm...as I said, all y'all doesn't make it plural, because y'all is usually already plural. And I beg to differ on your other claim. In TX, at least, y'all is frequently singular...which was initially baffling to a native Minnesotan. I kept looking around to see who all they were talking to!
But really, southern grammer is sort of an oxymoron anyway.