An example of a non-(locally cyclic) Abelian group whose automorphism group is cyclic not of order $2$

In the wake of my curiosity on this kind of things, I was thinking if there is an example of a non-(locally cyclic) Abelian group whose automorphism group is cyclic not of order $2$. Every example I know (of a non-cyclic group) with cyclic automorphism group lead me to $\mathbb{Z}_2$.

Is it known if such examples do exist?