For which values of $ n \geq 2 $ and $ q $ a prime power are all automorphisms of $ SL(n,q) $ induced by conjugation by elements of $ GL(n,q) $?
Old version of the question:
$ \Sigma L(n,q) $ is an almost quasisimple group defined to be the subgroup of $ GL(n,q) $ generated by the quasisimple group $ SL(n,q) $ together with all elements $ g \in GL(n,q) $ such that conjugation by $ g $ induces an outer autmorphism of $ SL(n,q) $. For which $ n,q $ is it the case that all outer automorphisms of $ SL(n,q) $ can be realized in this way?
(warning: the next couple sentences apparently contain several incorrect claims)
In other words for which $ n,q $ does $ \Sigma L(n,q) $ have the structure of $ \Sigma L(n,q) \cong SL(n,q).Out(SL(n,q)) $?
I know for example that this is true for $$ \Sigma L(2,5) \cong SL(2,5).Out(SL(2,5))=2.PSL(2,5).2 $$ an almost quasisimple group of order $ 240 $ which is index $ 2 $ in $ GL(2,5) $.
If $n > 2$ then the duality (graph) automorphism $A \mapsto (A^{-1})^{\mathsf T}$ is an outer automorphism of ${\rm SL}(n,q)$ that is not induced by conjugation in ${\rm GL}(n,q)$ (those are called diagonal automorphisms).
If $q$ is not a prime then there are field automorphisms of ${\rm SL}(n,q)$, which are not diagonal.
Conversely, for the groups ${\rm SL}(2,p)$ with $p$ prime, we claim that an automorphism $\alpha$ is induced by conjugation by an element of ${\rm GL}(2,p)$.The normalizer of a Sylow $p$-subgroup $P$ in ${\rm GL}(2,p)$ acts transitively on its non-identity elements, so we can assume that $\alpha$ centralizes $P$. But then, since a $p$-cycle in $S_{p+1}$ is self-centralizing, by considering the natural projective action we find that $\alpha$ is induced by conjugation by an element of $P$, which proves the claim.
So all automorphisms are diagonal if and only if $n=2$ and $q$ is prime.
In fact all this can be deduced directly from the standard classification of the automorphism groups of simple groups of Lie type, which are generated by diagonal, field, and graph automorphisms.