An Example of Outer Automorphism of $S_6$ with order 2?

Isaacs's treatment is memorable for me. $S_5$ has $6$ Sylow $5$-subgroups. The conjugation action here yields a homomorphism $S_5 \rightarrow S_6$ which has to be injective (noting $A_5$ is an impossibly large kernel for this action). This is already strange, as we have obtained a subgroup $H \leq S_6$ isomorphic to $S_5$, acting transitively on the 6 letters.

Next, consider the action of $S_6$ on the left cosets of $H$. This action gives a homomorphism $\sigma: S_6 \rightarrow S_6$ which also has to be injective, hence bijective. Here, the inverse $\sigma^{-1}$ maps the stabilizer of a single letter (one of the "usual" $S_5$'s in $S_6$) to the subgroup $H$ which is transitive. But conjugation takes a point stabilizer to another. Hence $\sigma ^{-1}$ can't be an inner automorphism.

I am not sure if it is easy to see that $\sigma^2$ is inner. The inner automorphisms having index $ \leq 2$ inside the full automorphism group should be doable the same way one shows there are no outer automorphisms in other symmetric groups, with a conjugacy class counting.