The regular and profinite representations of residually finite groups

Let $G$ be a residually finite group. To any decreasing sequence $\mathcal S = (H_n)_n $offiniteindexsubgroupsof$G$ is associated a unitary representation ${\rho }_{\mathcal{S}}$ of $G$ in the Hilbert space ${⨁}_{n=0}^{+\mathrm{\infty }}{\mathrm{\ell }}^2(G\mathrm{/}{H}_n)$. This paper investigates the following question: when does the representation ${\rho }_{\mathcal{S}}$ weakly contain the regular representation $\lambda$ of $G$?