Can a player trade 3 groups in one turn?
On Page 7 of the rule book, it says:


  Each Group exchanged counts as an action
  for the player whose turn it is, whichever way the Group
  moves. So if you trade a Group for a Group on your turn,
  that counts as two actions – one to move the first Group, one
  to move the second one!


Then, on Page 8, it says:


  When a Group is transferred to another player, its puppets (if any)
  must go with it, along with all treasuries involved. The player
  receiving the Group must immediately fit it, and its puppets, into his
  power structure...


(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.)