We can define the (first) homology of a surface $S$ by working with graphs embedded in $S$. That is, we take any (oriented) graph which is 2-cell embedded in $S$, and take cycles modulo boundaries in the usual way. Here, I am talking about homology with coefficients in $\mathbb{Z}$. The group that we get is independent of the graph, so is indeed a topological invariant of the surface.

I work with group-labelled graphs, which are oriented graphs with their edges labelled from a finite abelian group $\Gamma$. Proceeding as above, group-labelled graphs allow us to define group-labelled surfaces. That is, let $G$ be a $\Gamma$-labelled graph 2-cell embedded in a surface $S$. If each face of the embedding has group-value zero (the labels of edges on the boundary of the face sum to zero), then this gives a well-defined map on homology. In fact, the embedding of $G$ in S induces a homomorphism from $H_1(S)$ to $\Gamma$. So, we can forget about the $\Gamma$-labelled graph and just study this homomorphism.

My question is: how does this construction relate to taking homology with coefficients from $\Gamma$?

Someone once told me that what I am really doing is working with cohomology with coefficients in $\Gamma$, but I didn't really get this. Can someone please clarify?