Lifting images of standard representations of symmetric groups

We investigate closed subgroups $G\subseteq {\mathrm{S}\mathrm{p}}_{2g}({\mathbb{Z}}_2)$ whose modulo-$2$ images coincide with the image $\mathfrak{S}{2g + 1} \subseteq $\mathrm{S}\mathrm{p}${2g}($\mathbb{F}$2)$of$S{2g + 1}$ortheimage$\mathfrak{S}{2g + 2} \subseteq $\mathrm{S}\mathrm{p}${2g}($\mathbb{F}$2)$of$S{2g + 2}$underthestandardrepresentation\mathrm{.}Weshowthatwhen$g \geq 2$, the only closed subgroup $G\subseteq {\mathrm{S}\mathrm{p}}_{2g}({\mathbb{Z}}_2)$ surjecting onto ${\mathfrak{S}}_{2g+2}$ is its full inverse image in ${\mathrm{S}\mathrm{p}}_{2g}({\mathbb{Z}}_2)$, while all subgroups $G \subseteq $\mathrm{S}\mathrm{p}$_{2g}($\mathbb{Z}$2)$surjectingonto$\mathfrak{S}{2g + 1}$ are open and contain the level-$8$ principal congruence subgroup of ${\mathrm{S}\mathrm{p}}_{2g}({\mathbb{Z}}_2)$. As an immediate application, we are able to strengthen a result of Zarhin on $2$-adic Galois representations associated to hyperelliptic curves. We also prove an elementary corollary concerning even-degree polynomials with full Galois group.