I have a proof for the following assertion which employs Model Theory. It has certainly a pure group theoretic proof, but what is such a proof? Is the assertion trivial?

**Theorem**  Let $G$ be a finite group and $s\in G$ be an arbitrary element. Suppose $A=C_{\mathrm{Aut}(G)}(s)$. Then $C_G(A)$ is cyclic.

Edition: By the counterexample of Khalid, it seems that the correct statement is following:

**Theorem**  Let $G$ be a finite group and $s\in G$ be an arbitrary element. Suppose $A=C_{\mathrm{Aut}(G)}(s)$ has odd order. Then $C_G(A)$ is cyclic.