  If G is a semidirect product N by H with N normal and finitely generated then
G has the property that every finite group is a quotient of some finite index
subgroup of G if and only if one of N and H has this property. This has
applications to 3-manifolds and to cyclically presented groups, for instance
for any fibred hyperbolic 3-manifold M and any finite simple group S, there is
a cyclic cover of M whose fundamental group surjects to S. We also give a short
proof of the residual finiteness of ascending HNN extensions of finite rank
free groups when the induced map on homology is injective.
