Coprime action of simple groups

Question

Let $A$ act on $G$ coprimely by automorphism where $G$ is a nonabelian simple group. Does it imply that $|A|=p$? (where $p$ is a prime number)

If not, Is there any source that examines such cases separately and gives the cardinality of $C_G(A)$?

Answer 1

Assuming that acting coprimely means $(|A|,|G|)=1$ and that $p$ is supposed to be a prime, I think the answer is no. I think (although I haven't checked carefully) that the only coprime automorphisms of finite simple groups are field automorphisms of groups of Lie type.

For example ${\rm PSL}(2,2^{25})$ has order coprime to $5$ and has a field automorphism group of order $25$. Its centralizer is ${\rm PSL}(2,2)$.

And ${\rm PSL}(2,2^{35})$ has a field automorphism of order $35$ to which it is coprime.