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?

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    $\begingroup$ @FedorPetrov I don't understand your comment. The answer to the question is no because $C_6$ is a counterexample. $\endgroup$ – Derek Holt Apr 8 '17 at 8:09
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    $\begingroup$ More generally, if $p$ and $q$ are distinct primes then $C_{pq}$ yields a counterexample. $\endgroup$ – Francesco Polizzi Apr 8 '17 at 8:17
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    $\begingroup$ No. There exists plenty of non-cyclic $p$-groups of exponent $p$ whose automorphism group is a $p$-group as well (say, assuming that $p$ is larger than the nilpotency length, coming from a Lie algebra over the field with $p$ elements with unipotent automorphism group). [I leave this as a comment, since you will soon once again change the question to discard such examples.] $\endgroup$ – YCor Apr 8 '17 at 14:53
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    $\begingroup$ @YCor - Unless you posted an answer giving the cyclic counterexample which you subsequently deleted, I think the bracketed comment is unnecessarily ungenerous toward the OP. I understand your annoyance at the edit that moved the goalposts. But (a) the OP is a low-rep and thus probably inexperienced user so may not have a refined sense of the norms around this; (b) to my knowledge nobody has posted an answer so 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. $\endgroup$ – benblumsmith Apr 8 '17 at 20:15
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    $\begingroup$ @benblumsmith have you been through the history of questions asked by the OP, their comments, and their edits? $\endgroup$ – YCor Apr 8 '17 at 20:31

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