Let $S$ be a finite nonabelian simple group such that the exact sequence

$$1 \to S \to {\rm Aut}(S) \to {\rm Out}(S) \to 1$$

is nonsplit, where $S$ is identified with ${\rm Inn}(S)$. Then there is always a (not necessarily unique) minimal subgroup $A$ in ${\rm Out}(S)$ with respect to the condition that

$$1 \to S \to S.A \to A \to 1$$

is nonsplit. Where can I find a classification of all such pairs $(S,A)$? In particular, is it always true that $|A|=2$? I checked it with the ATLAS that $S$ cannot be sporadic and, if it is alternating, then $S\cong A_6$ and $S.A\cong M_{10}$.

Update: (inspired by Derek Holt's example below)

Can there be nonisomorphic minimal nonsplit extensions $S.A_1$ and $S.A_2$ for a given simple group $S$?