Hallo!

I'm looking for a reference. I'm sure that the information I need is already in the literature but I'm having some trouble to find it. Here is the question.

Let $S$ be a non-abelian finite simple group (the only case I'm really interested in is for groups of Lie type) and let $A$ be the automorphism group of $S$. For which primes $p$, every element of order $p$ in $S$ has $C_A(x)$ (the centralizer of $x$ in $A$) abelian?

Typically, one might think that $p$ is a primitive prime divisor of $q^n-1$ (where $q$ is the size of the defining field of $S$ and $n$ is, roughly, its Lie rank). However, already for these elements I'm not able to find any reference for $C_A(x)$ (despite the fact that lots is known on $C_S(x)$).