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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?

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  • $\begingroup$ Take $G=A_n$, where $n$ is even (but not $6$). Then $A=S_n$, and if we take $\alpha=(1,2, \ldots, n)$, then $C_G(\alpha)=\langle\alpha^2\rangle$, which has order $n/2$. As $n\rightarrow\infty$, this becomes a very small proportion of $G$. $\endgroup$ – Steve D Nov 1 '17 at 17:22

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