Can groups of automorphism over non-isomorphic groups be isomorphic?

Question

Can two groups of automorphisms over non-isomorphic groups be isomorphic? If $G$ and $G'$ are non-isomorphic groups and $\text{Aut}(G)$ and $\text{Aut}(G')$ be their group of automorphisms, then can they be isomorphic to each other or not?

Answer 1

Aut$(\Bbb Z_3)$ and Aut$(\Bbb Z_4)$ are both isomorphic to $\Bbb Z_2$.

Answer 2

A non-abelian example, $Aut(Q) \cong S_4 \cong Aut(S_4)$. Here $Q$ is the quaternion group of $8$ elements.

Answer 3

As a more complicated example, we have $\operatorname{Aut}(C_2\times C_8)\cong C_2\times D_8$ and $$\operatorname{Aut}(C_2\times C_{12})\cong\operatorname{Aut}(C_3)\times\operatorname{Aut}(C_2\times C_4)=C_2\times D_8.$$