"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.