  We present a method for computing the number of epimorphisms from a
finitely-presented group G to a finite solvable group \Gamma, which generalizes
a formula of G\"aschutz. Key to this approach are the degree 1 and 2 cohomology
groups of G, with certain twisted coefficients. As an application, we count
low-index subgroups of G. We also investigate the finite solvable quotients of
the Baumslag-Solitar groups, the Baumslag parafree groups, and the Artin braid
groups.
