  We discuss how to apply the dimer method to Ising models on group lattices
having non trivial topological genus $g$. We find that the use of group
extension and the existence of both external and internal group isomorphisms
greatly reduces the number of distinct Pfaffians and leads to explicit
topological formulas for their sign and weight in the expansion of the
partition function. The complete solution for the Ising model on the Klein
lattice group $L(2,7)$ with $g=3$ is given.
