Closed orbits in quotient systems

We study the relationship between pairs of topological dynamical systems $(X,T)$ and $(X^{\mathrm{\prime }},T^{\mathrm{\prime }})$, where $(X^{\mathrm{\prime }},T^{\mathrm{\prime }})$ 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.