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](http://en.wikipedia.org/wiki/ATLAS_of_Finite_Groups) 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 $S$ with $A\_1\cong A\_2$?



Finite simple groups: smallest nonsplit automorphic extensions
It is not always true that $|A|=2$. The outer automorphism of $A\_6 \cong {\rm PSL}\_2(9)$ induced in $M\_{10}$ is the product of a field automorphism and a diagonal automorphism of ${\rm PSL}\_2(9)$ and I believe that it is true in general that the product of a field and a diagonal automorphism of the same order of a finite simple group of Lie type gives rise to a nonsplit extension. I checked this by computer in the case of $S = {\rm PSL}\_3(64)$, where we get a nonsplit extension $S.A$ with $|A|=3$.


I am guessing that these are essentially the only instances of nonsplit extensions of finite simple groups by their automorphism groups, but I am not certain, and I am afraid that I not aware of any definitive results of this type. It is an interesting question.


