Can a non-inner automorphism map every subgroup to its conjugate?

Question

Let $G$ be a finite non-cyclic group. Can a non-inner automorphism map every subgroup to its conjugate? Namely, can there be a non-inner automorphism $\alpha$ that, for every $H\le G$, there exists some $g$ in $G$ such that $\alpha(H)=H^g$?

Answer 1

Yes, the dihedral group of order 10 has this property. Its subgroup structure is very simple: $D_{10}$, $\{e\}$, the rotation subgroup, and the 5 subgroups generated by a flip. Any automorphism fixes the first three, so those are done, and shuffles the last five, and those 5 subgroups are all conjugate to each other.

All we need now is to show that there is a non-inner automorphism, but this is easy; the inner automorphisms always send a rotation to its inverse (or fix it) so we only need an automorphism which doesn't do that. Let the generators be $\sigma,\tau$, rotation and flip, and consider the automorphism defined on the rotations by $\sigma \mapsto \sigma^2$ and fixing $\tau$.