Monodromy groups of Hurwitz-type problems

We solve the Hurwitz monodromy problem for degree 4 covers. That is, the Hurwitz space H4,g of all simply branched covers of P1 of degree 4 and genus g is an unramified cover of the space P2g+6 of (2g+6)-tuples of distinct points in P1. We determine the monodromy of π1(P2g+6) on the points of the fiber. This turns out to be the same problem as the action of π1(P2g+6) on a certain local system of Z/2-vector spaces. We generalize our result by treating the analogous local system with Z/N coefficients, 3∤N, in place of Z/2. This in turn allows us to answer a question of Ellenberg concerning families of Galois covers of P1 with deck group 2(Z/N):S3.