Let $A_n$ be the alternating group of degree $n$, and $S$ be the set of all subgroups of $A_n$ with index $n$. Prove that the natural action of $\mathrm{Aut}(A_n)$ on $S$ is transitive.
This appears as an exercise problem in S. Lang's Algebra, where the result is then used to find the exotic structure of $\mathrm{Aut}(A_6)$. Therefore a proof should not involve too many deep insights on $A_n$. I think there is a direct way to attack(since this is only an exercise), but I am having a difficult time figuring out how a subgroup of $A_n$ with index $n$ looks like or how to desribe an automorphism of $A_n$ directly(i.e. without mentioning further results about $\mathrm{Aut}(A_n)$).