TITLE: Infinite noncyclic groups with abelian automorphism group
QUESTION [5 upvotes]: Is there any infinite noncyclic group whose automorphism group is abelian..can we find a sufficient condition for infinite group to have an abelian automorphism group
Thank you

REPLY [7 votes]: The finite abelian groups that can be the automorphism group of an infinite abelian group have been classified by Fournelle in [Finite groups of automorphisms of infinite groups II, J. of Algebra 80, 1983, 106 - 112, Theorem 1.2]: 

There is an infinite abelian group $A$ with $Aut(A) = G$ for a finite abelian group $G$ iff $G$ is of even order and is a direct product of cyclic groups of orders 2, 3, and 4 with the property that if $G$ has an element of order 12 it also has an element of order 2 that is not a sixth power. 

Examples of torsion-free groups $A$ with $Aut(A)=G$ for $G$ as above are constructed in [Fuchs: Infinite abelian groups II, Chap. XVI, Sect. 116] in  examples 1, 2 and Theorem 116.2.