In introduction to algebra we got the exercise:
Let $G$ be a group. Show that when $\operatorname{Aut}(G)$ is cyclic $G$ is abelian.
This doesn't make that much trouble. Denote the center (all commuting elements) with $Z$. Then $G/Z$ is isomorphic to $\operatorname{Int}(G)$ where $\operatorname{Int}(G)$ denotes the subgroup of inner automorphisms. As every subgroup of a cyclic group is cyclic we have $G/Z$ is cyclic and hence $G$ is abelian (As a group which is cyclic of the center is abelian).
So the question is:
Is there a non cyclic group with a cyclic automorphism group?
We discussed the question already in chat and I found (thanks to google) a solution there, but I would enjoy other Examples.