  We solve the Hurwitz monodromy problem for degree-4 covers. That is, the
Hurwitz space H_{4,g} of all simply branched covers of P^1 of degree 4 and
genus g is an unramified cover of the space P_{2g+6} of (2g+6)-tuples of
distinct points in P^1. We determine the monodromy of pi_1(P_{2g+6}) on the
points of the fiber. This turns out to be the same problem as the action of
pi_1(P_{2g+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,
gcd(3,N)=1, in place of Z/2. This in turn allows us to answer a question of
Ellenberg concerning families of Galois covers of P^1 with deck group
(Z/N)^2:S_3.
