Abelian automorphism group not implies cyclic

Statement
It is possible to have a group whose automorphism group is abelian but such that the group is not a cyclic group.

Opposite facts

 * Finite abelian and abelian automorphism group implies cyclic

Facts used

 * 1) uses::Locally cyclic implies abelian automorphism group, uses::locally cyclic not implies cyclic
 * 2) uses::Abelian and abelian automorphism group not implies locally cyclic
 * 3) uses::Abelian automorphism group not implies abelian, uses::cyclic implies abelian

Proof
We can counterexamples generated from Fact (1), (2), or (3).