  In this paper we obtain realizations of the 4-dimensional general symplectic
group over a prime field of characteristic $\ell>3$ as the Galois group of a
tamely ramified Galois extension of $\mathbb{Q}$. The strategy is to consider
the Galois representation $\rho_{\ell}$ attached to the Tate module at $\ell$
of a suitable abelian surface. We need to choose the abelian varieties
carefully in order to ensure that the image of $\rho_{\ell}$ is large and
simultaneously maintain a control on the ramification of the corresponding
Galois extension. We obtain an explicit family of curves of genus 2 such that
the Galois representation attached to the $\ell$-torsion points of their
Jacobian varieties provide tame Galois realizations of the desired symplectic
groups.
