Suppose $\Gamma_1\leq \Gamma_2$ are two finite groups, and $M$ a smooth manifold such that $M/\Gamma_i$ is again a smooth manifold.
I want to know
Qusetion1: is there any relation between these two quotient spaces $M/\Gamma_1$ and $M/\Gamma_2$?
AND Is that possible for two distinct finite groups $\Gamma_1, \Gamma_2$ (with same order and non-isomorphic or of completely different order and $\Gamma_i\not\leq \Gamma_j$) produce homeomorphic or diffeomorphic quotient spaces $M/\Gamma_1\simeq M/\Gamma_2$? (I work in category of Smooth manifolds.)
In dim=2 I know real projective plane that is $\Bbb Z_2$ action on sphere but I don't know other one. An explicit example may clear the problem to me.
A less-related post without answer: Possible manifold quotients of spheres by finite isometric actions