Cyclic automorphism group not implies cyclic

Statement
It is possible to have a group that is an [group whose automorphism group is cyclic]] (i.e., the automorphism group is a cyclic group) but the group itself is not a cyclic group.

Related facts

 * Finite and cyclic automorphism group implies cyclic
 * Cyclic automorphism group implies abelian
 * Finite abelian and abelian automorphism group implies cyclic
 * Abelian and abelian automorphism group not implies locally cyclic

Proof
Let $$G$$ be the group of rational numbers with square-free denominators, i.e., $$G$$ is the subgroup of the additive group of rational numbers comprising those rational numbers that, when written in reduced form, have denominators that are square-free numbers, i.e., there is no prime number $$p$$ for which $$p^2$$ divides the denominator. Then:


 * The only non-identity automorphism of $$G$$ is the negation map, so the automorphism group is cyclic group:Z2, and is hence cyclic.
 * $$G$$ is not a cyclic group. In fact, it is not even a finitely generated group because any finite subset of $$G$$ can only cover finitely many primes in their denominators. It is, however, a locally cyclic group: any finitely generated subgroup is cyclic.