  We consider a two-sided Pompeiu type problem for a discrete group $G$. We
give necessary and sufficient conditions for a finite set $K$ of $G$ to have
the $\mathcal{F}(G)$-Pompeiu property. Using group von Neumann algebra
techniques, we give necessary and sufficient conditions for $G$ to be a
$\ell^2(G)$-Pompeiu group
