The two-sided Pompeiu problem for discrete groups

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