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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?

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"Groups with a small number of automorphisms" by H. de Vries, A. B. de Miranda (Math Zeitschrift (1957/58) Volume 68, Issue 1, pp 450-464) link gives examples with cyclic automorphism groups of order 4 and 6 (and it looks as though 2,4 and 6 may be the only possible orders, but I'm not completely sure about that).

Volume 2 of "Infinite Abelian Groups" by L. Fuchs has chapters with lots of information about endomorphism rings and automorphism groups of abelian groups, which may interest you.

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