Two groups that are the automorphism groups of each other

Solution 1:

An answer in the negative for this question would immediately prove the Smith conjecture: Does the finite part of the automorphism tower of a group G eventually repeat? (Smith 1964)

I have spent some time wrestling with this conjecture, and the pattern I observed was that this did occur and typically very rapidly (only a few steps in, I was using GAP to do the computations, as order of the groups also typically grew rapidly), moreover, every time I found a group for which it began repeating, it was also a reptend of length 1 (meaning the 'last' group was isomorphic to its automorphism group).

If the Smith conjecture is true, and reptend is always length 1, then your question's answer is false, otherwise if the Smith conjecture is true, so is your conjecture, but if Smith's conjecture is false, then it's unclear whether or not you conjecture is true or false.