Zarhin proves that if
$C$ is the curve
${{y}^{2}}\,=\,f(x)$ where
$\text{Ga}{{\text{l}}_{\mathbb{Q}}}(f(x))\,=\,{{S}_{n}}$ or
${{A}_{n}}$, then
$\text{En}{{\text{d}}_{\overline{\mathbb{Q}}}}(J)\,=\,\mathbb{Z}$. In seeking to examine his result in the genus
$g\,=\,2$ case supposing other Galois groups, we calculate
$\text{En}{{\text{d}}_{\overline{\mathbb{Q}}}}(J)\,{{\otimes }_{\mathbb{Z}}}\,{{\mathbb{F}}_{2}}$ for a genus 2 curve where
$f(x)$ is irreducible. In particular, we show that unless the Galois group is
${{S}_{5}}$ or
${{A}_{5}}$, the Galois group does not determine
$\text{En}{{\text{d}}_{\overline{\mathbb{Q}}}}(J)$.