Rules specify that when trading each group exchanged counts as an action to the player whose turn it is. And then later goes on to say that puppets of a group are transferred along with it. Is the player then limited to exchanging 1 group for another group or two groups for cash/special cards? Or can he exchange a parent group with 3 puppets for another group?
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On Page 7 of the rule book, it says:
Then, on Page 8, it says:
(I know you already pointed that out, I just wanted to show the exact rules quotes.)
That would lead me to believe that you can trade off one of your own Groups (and any of it's puppets) for another Group (and any of it's puppets). Doing so will cost your both of your actions for the turn, one to 'release' the old Group and another to 'welcome' the new Group in. The puppets don't have a say in where they are going, so so they don't count as additional 'Groups' in this case.
So you are fine to trade away a parent Group with three puppets for another group, just as long as you get a good enough deal!
(Note, that if a player can't fit a traded group and all it's puppets into their Power Structure, some of the puppets may have to be released to the uncontrolled area.)