  We study the relationship between pairs of topological dynamical systems $
(X,T) $ and $ (X',T') $, where $ (X',T') $ is the quotient of $ (X,T) $ under
the action of a finite group $ G $. We describe three phenomena concerning the
behaviour of closed orbits in the quotient system, and the constraints given by
these phenomena. We find upper and lower bounds for the extremal behaviour of
closed orbits in the quotient system in terms of properties of $ G $ and show
that any growth rate in between these bounds can be achieved.
