The image of the mod-$\ell$ Galois representation associated to a genus 2 curve is maximal if it is as large as possible given constraints imposed by the Weil pairing and endomorphisms of the Jacobian.

For generic curves the geometric endomorphism ring of the Jacobian is $\Z$ and the maximal image is $\GSp(4,\Z/\ell\Z)$. In all other cases it is a proper subgroup of $\GSp(4,\Z/\ell\Z)$.