For a hyperelliptic curve H, the mod 2 monodromy is smaller than $GSp_{2g}(F_2)$ -- since the two torsion of the Jacobian H is generated by differences of Weierstrass point, the monodromy of a generic curve is the symmetric group.

The degree 3 map $C \to \mathbb{P}^1$ associated to a trigonal curve may only have ramification of type (2,1), and thus may not give any "easy to see" 3 torsion points on the Jacobian. So it is reasonable to ask if a generic such curve has maximal mod l monodromy for every l.