Let $G$ be a finite centerless group, which we identify with $\operatorname{Inn}(G)$, and note $A := \operatorname{Aut}(G)$. Suppose further that $[A:G] = 2$, and take a non-inner automorphism $\alpha \in A$.
What can we say about the $C_G(\alpha) = C_A(\alpha) \cap G$, i.e. the group of elements which are fixed by the action of $\alpha$? Clearly $\alpha$ must move some elements, lest it be the identity, but is there any condition on the (number of) elements that it can fix?