In Scott's classic textbook on Group Theory, he asks:

Suppose that $G$ is a finite group. Is the sequence of isomorphism types
of the groups $Aut^{(n)}(G)$ for $n \in \mathbb{N}$ eventually periodic?

Here $Aut^{(2)}(G) = Aut(Aut(G))$ etc. Equivalently, is the sequence
$|Aut^{(n)}(G)|$ always bounded above?

It apparently remains opens whether the sequence of automorphism types of $Aut^{(n)}(G)$ is in fact always eventually constant. So I would like to ask:

Does there exists a finite group such that $Aut(G) \not \cong G$ but
$Aut^{(n)}(G) \cong G$ for some $n \geq 2$?