If G is an almost simple group, then Aut(G) is complete?


Apologies - I meant to post this on Stack Exchange


Just wondering if anyone has a reference to the above - it's quoted on Wikipedia (so supposing that means it's true...) and seems a plausible result.


I've tried generalising the proof of the version when G is simple. The proof I've seen for a simple group G relies on $Inn(G)$ char $Aut(G)$, which I'm not sure holds in the almost simple case?


I'm presuming there is a CFSG free proof.



If G is an almost simple group, then Aut(G) is complete?
This isn't true. Let $S$ be a simple group such as ${\rm PSL}(4,5)$ with outer automorphism group isomorphic to $D\_8$ (dihedral of order $8$), and let $G = S.2$ be an extension of $S$ by a non-central involution in $D\_8$.


Then ${\rm Aut}(G) = S.2^2$ and ${\rm Aut}({\rm Aut}(G)) = S.D\_8 = {\rm Aut}(S)$.


