Let $G$ be a finite group with no element of order $p^2$ for each prime $p$. Also, suppose that $\vert G\vert\neq p$, for each prime $p$. Does there always exist an automorphism $\phi$ of order 2 such that for at least one subgroup of $G$, say $H$, we have $\phi(H)\neq H$?

Update: What if we add the supposition that $G$ is not cyclic?

answerso the update did not invalidate an actual answer, and (c) at least the OP (with the word "update") explicitly acknowledged the edit, so readers can see the history. $\endgroup$7more comments