Reconstructing a set from its subset sums: $2$-torsion-free groups

For a finite multiset $A$ of an abelian group $G$, let $\text{FS}(A)$ denote the multiset of the $2^{\mathrm{\mid }A\mathrm{\mid }}$ subset sums of $A$. It is natural to ask to what extent $A$ can be reconstructed from $\text{FS}(A)$. We fully solve this problem for $2$-torsion-free groups $G$ by giving characterizations, both algebraic and combinatorial, of the fibers of $\text{FS}$. Equivalently, we characterize all pairs of multisets $A,B$ with $\text{FS}(A)=\text{FS}(B)$. Our results build on recent work of Ciprietti and the first author.