Consider the following situation: We have a countable discrete group $G$ with a finite index (not necessarily normal) subgroup $H$. It is possible to construct 3 $CW$ complexes and a covering diagram as usual. [I am skipping the details, see for instance Geoghegan's Top. Methods in Group Theory]
By considering the free $R$ modules generated by the cosets of $H$ in $G$ it is possible to relate the cellular homology of these spaces(with coefficients) with the homologies of the groups. [please again see the above book.]
Now, the question: Suppose $f\colon H\rightarrow G$ is a homomorphism. How does this $f$ enters the above story? For example, the inclusion homomorphism $\subset \colon H\subset G$ (if I don't lie to myself) appears as the convering of complexes. Does a general homomorphism have a topological meaning?