It is well-known that the map $\operatorname{Aut}$ from the class of groups to itself has fixed points. For $n \neq 2$ or $6$, $\operatorname{Aut}(S_n) \cong S_n$, $\operatorname{Aut}(D_4) \cong D_4$ and if $G$ is a finite non-abelian simple group, then $\operatorname{Aut}(\operatorname{Aut}(G)) \cong \operatorname{Aut}(G)$.
The question arises if there is a fixed point among the non-trivial abelian groups. It is relatively easy to see that finitely generated abelian groups do not qualify, so if such a group exists, it has to be infinitely generated. Note that for topological automorphisms of topological groups fixed points are known: for instance $\operatorname{Aut}(\mathbb{R} \times \mathbb{Z}/2\mathbb{Z}) \cong \mathbb{R} \times \mathbb{Z}/2\mathbb{Z}$. But we assume no additional structure imposed.
The automorphism group, as an additive group, of the $p$-adic integers $\mathbb{Z}_p$ is the same as its group of units, which is isomorphic to $\mathbb{Z}_p\times\mathbb{Z}/(p-1)\mathbb{Z}$ for odd primes $p$. So $\mathbb{Z}_3\times\mathbb{Z}/2\mathbb{Z}$ is isomorphic to its own automorphism group.