TITLE: Two groups that are the automorphism groups of each other
QUESTION [33 upvotes]: Let $H,K$ be two non-isomorphic  groups such that $H\cong Aut(K)$ and $K\cong Aut(H)$.
Is there any example of such groups ?
Note: I had asked the question there.

REPLY [6 votes]: Computer search in sage/gap didn't found any solutions for orders up to $120$.
It didn't assume the orders are equal.