Let $G$ be an affine algebraic group defined over a field of charactersitic $0$. Let $H$ be a finite subgroup of $G$. Left translations by geometric points of $H$ yield a finite group of automorphisms on $G$, and thus $G/H$ can be defined as an affine variety. Is the morphism $G\rightarrow G/H$ étale?
Both $G$ and $G/H$ are nonsingular, and fibers over geometric points are all isomorphic to $H$, so this suggests to me a higher level of regularity than the general case of quotients by finite groups of automorphisms.