Let $S$ be a connected locally noetherian scheme with $s$ a geometric point of $S$. I read something like this: to give a surjective continuous homomorphism from $\pi_1(S, \bar{s})$ to a finite group $G$ is equivalent to give a connected finite etale covering $X\to S$ equipped with a simply transitive action of $G$ on the fibre $X(\bar{s})$.
I don't really need a rigorous proof of this fact. But I would like know the correspondence. Given a a surjective continuous homomorphism $\pi_1(S,\bar{s})\to G$. Then the kernel of this homomorphism, say $H$, is a normal open subgroup of $\pi_1(S,\bar{s})$ of finite index. Is $H$ the fundamental group of some (connected) finite etale cover $X$ of $S$? Is the quotient $\pi_1(S,\bar{s})/H$ the automorphism group $\mathrm{Aut}(X/S)$ of $X$.
On the other hand, given a connected finite etale cover $X\to S$, and a simply transitive action of a finite group $G$ on $X(\bar{s})$, is $G$ isomorphic to $\mathrm{Aut}(X/S)$? This does not seem very true to me.