Maps of surface groups to finite groups with no simple loops in the kernel

Let F-g denote the closed orientable surface of genus g. What is the least order finite group, G(g), for which there is a homomorphism psi: pi (1)(F-g) --> G(g) so that no nontrivial simple closed curve on F-g represents an element in Ker(psi)? For the torus it is easily seen that G(1) = Z(2) x Z(2) suffices. We prove here that G(2) is a group of order 32 and that an upper bound for the order of G(g) is given by g(2g+1). The previously known upper bound was greater than 2(g22g).