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?



An example of a non-(locally cyclic) Abelian group whose automorphism group is cyclic not of order $2$
"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](http://link.springer.com/content/pdf/10.1007/BF01160361.pdf)
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.


