I have wondered what these public groups could be from time to time and it is a feature that can be employed by administrators of vbulletin forums, or so I gather from attempting a little research. They are not employed in these forums (unless I am missing something) ..so that is why you see no members belonging to public groups. It is nothing to do with groups unrelated to the Forums, and can in any event only be edited, I would assume, by forum administrators. You now know as much as I do, and are probably utterly mistified ...as I am
all I can say, is take no notice