Suppose $G$ is a split semisimple $\mathbf{Q}$-group and $\Gamma \subset G(\mathbf{Q})$ is a lattice. Conjectures due to Ash and his collaborators (elaborating on earlier work of Serre) predict a fairly precise correspondence between continuous representations $\rho: \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to\widehat{G}(\overline{\mathbf{F}}_p)$ and "Hecke eigenclasses" in $H^{\ast}(\Gamma,\overline{\mathbf{F}}_p)$. See for example this paper where the conjecture is elaborated very precisely for $\mathrm{GL}_n/\mathbf{Q}$, and this paper for a more general prediction.

The really remarkable thing here is that for may groups $G$ - say, if $G(\mathbf{R})$ does not admit discrete series - there should be a serious paucity of non-torsion characteristic zero homology, and the classes predicted by Galois representations will often not be the mod-$p$ reduction of some characteristic zero class! So these genuine torsion classes should be tied rather intimately to Galois representations - that seems pretty remarkable to me!