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 $\mathrm{\Pi }:\mathsf{E}\mathsf{x}\mathsf{i}\mathsf{t}(M)\to \mathsf{E}\mathsf{x}\mathsf{i}\mathsf{t}(M\mathrm{/}G)$ is a right fibration and $\mathsf{E}\mathsf{n}\mathsf{t}\mathsf{e}\mathsf{r}(M\mathrm{/}G)$ is classified by a natural functor $\mathsf{E}\mathsf{n}\mathsf{t}\mathsf{e}\mathsf{r}(M\mathrm{/}G)\to O_G$, 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 $M\mathrm{/}G$ to homotopies in $M$.