Exit path categories induced by group actions

We prove a structural result concerning the exit path category associated to a manifold $M$ equipped with a smooth action of a finite group $G$. Specifically, the functor $\Pi: \mathsf{Exit}(M) \rightarrow \mathsf{Exit}(M/G)$ is a right fibration and $\mathsf{Enter}(M/G)$ is classified by a natural functor $\mathsf{Enter}(M/G) \rightarrow O_G$, where $O_G$ is the orbit category of $G$. The main technical result manipulates exit paths to immediately exiting paths, enabling lifts of homotopies in $M/G$ to homotopies in $M$.