Is there an Abelian group $A$ which is not locally cyclic whose automorphism group is cyclic ?
Warning: This answer is not correct (see comments)!
Let $E$ be an elliptic curve over the complex numbers (to make things easy). $E$ is an abelian group which is not locally cyclic. For example, it contains the group $\mathbb Z \oplus \mathbb Z$ which is not locally cyclic.
Then it can be shown that if the j-invariant is neither $0$ or $1728$, then $Aut(E)\simeq \mathbb Z_2$, which is cyclic. The reference is this article.