Group isomorphic to its automorphism group
Solution 1:
The infinite dihedral group $G=D_{\infty}$ is also isomorphic to its own automorphism group, but is not complete - see the article A Note on Groups with Just-Infinite Automorphism Groups.
For finite groups, not $p$-groups, not complete, it is said here: "Whether there exist other groups isomorphic to their automorphism groups is an open problem". For more results, see the article On a question about automorphisms of finite p-groups by G. Cutolo. See also this MO-question.