In a 1952 paper M. Hall proved that if $G=\{a_1,\ldots,a_n\}$ is an additive abelian group of order $n$ and $b_1,\ldots,b_n$ are elements of $G$ with $b_1+\ldots+b_n=0$ then we have $$\{a_{\sigma(i)}+b_i:\ i=1,\ldots,n\}=\{a_1,\ldots,a_n\}$$ for some $\sigma\in S_n$.

Motivated by this, I raise the following question.

**Question.** Let $G$ be an additive abelian group, and suppose that $A$ is a nonempty subset of $G$ with $A-A=\{a-a':\ a,a'\in A\}$ equal to $G$. If $b_1,\ldots,b_n\in G$ with $b_1+\ldots+b_n=0$, whether there are $a_1,\ldots,a_n\in A$ and a permutation $\sigma\in S_n$ such that
$b_i=a_i-a_{\sigma(i)}$ for all $i=1,\ldots,n$?

I guess that this question has a positive answer. Your comments are welcome!