2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71089We present a method for computing the number of epimorphisms from a finitely-presented group G to a finite solvable group Γ, which generalizes a formula of Gäschutz. 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.30 pages; accepted for publication in the Journal of AlgebraGroup TheoryGeometric Topology20J05, 20F16; 20E07, 20F36, 57M05text