# Two groups that are the automorphism groups of each other

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 ?

• If the initial group is finite and centerless then there are no finite cycles of length more than $1$. – Pablo Jan 10 '15 at 23:04
Computer search in sage/gap didn't found any solutions for orders up to $120$.