If $\operatorname{Aut} (G)$ is isomorphic to $\operatorname{Aut} (H)$ then is it necessary that $G$ is isomorphic to $H$?
My answer is no. $\operatorname{Aut} (\mathbb{Z)}$ is isomorphic to $Z_2$ and $\operatorname{Aut} (Z_3)$ is also isomorphic to $U(3)$, which is isomorphic to $Z_2$. But $\mathbb Z$ is not isomorphic to $Z_3.$ Correct? Thanks