Let $G$ be a finite group with no element of order $p^2$ 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 about if add the supposition that $G$ is not cyclic of prime order?
No, take for example the cyclic group $C_2$. It certainly has no element of order $p^2$. Nevertheless, every automorphism is the identity, i.e., $\phi(H)=H$. So there exists no automorphism of order $2$ with $\phi(H)\neq H$ for some $H$.
A quick counter-example is any cyclic group $C_p$ of prime order $p>2$. There's no element of order $p^2$ and for any automorphism $\phi$, we have $\phi(\{1\})=\{1\}$ and $\phi(C_p)=C_p$ so there is certainly no automorphism of order $2$ with subgroup $H$ of $G$ with $\phi(H)\ne H$