What does the automorphism group of a free abelian group A know about A?

Let A be an infinitely generated free abelian group. We prove that the automorphism group Aut(A) first-order interprets the full second-order theory of the set vertical bar A vertical bar with no structure. In particular, this implies that the automorphism groups of two infinitely generated free abelian groups A(1), A(2) are elementarily equivalent if and only if the sets vertical bar A(1)vertical bar, vertical bar A(2)vertical bar are second-order equivalent.