I'd like to have a better understanding of this old valuable answer (I've tried to comment the post, but with no feedback).
I'm reading (correctly?):
- the first two Lemmata as: $\operatorname{Aut}(S_n)/\operatorname{Inn}(S_n)\cong\phi(\operatorname{Aut}(S_n))$, where $\phi$ is the action of $\operatorname{Aut}(S_n)$ on the set of the conjugacy classes of $S_n$;
- the third Lemma as: for $n\ne 6$, there's no class of involutions with the same size of the class of transpositions.
But how does from this follow that every automorphism of $S_{n\ne 6}$ is class-preserving (namely that for $n\ne 6$ the action is trivial)? For instance, $S_7$ has two conjugacy classes of involutions which have also the same size ($105$): how, from the three Lemmata, can one rule out the existence of some (outer) automorphism swapping such two classes? I feel like I'm misunderstanding (or not properly valorizing) something, so that I don't see how the Theorem follows.
As per the comments, now I see that, w.r.t. the action of $\operatorname{Aut}(S_n)$ on the set of conjugacy classes of $S_n$:
- from the second lemma, follows that $\operatorname{Stab}(\operatorname{cl}((ij)))=\operatorname{Inn}(S_n)$;
- from the third lemma, follows that, for $n\ne 6$, $\operatorname{Stab}(\operatorname{cl}((ij)))=\operatorname{Aut}(S_n)$.
Therefore, for $n\ne 6$, $\operatorname{Aut}(S_n)=\operatorname{Inn}(S_n)$. Thanks to @runway44 for their comments.