  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 $|A|$ 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 $|A_1|,|A_2|$ are
second-order equivalent.
