Lifting images of standard representations of symmetric groups

We investigate closed subgroups $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ whose modulo-$2$ images coincide with the image $\mathfrak{S}_{2g + 1} \subseteq \mathrm{Sp}_{2g}(\mathbb{F}_2)$ of $S_{2g + 1}$ or the image $\mathfrak{S}_{2g + 2} \subseteq \mathrm{Sp}_{2g}(\mathbb{F}_2)$ of $S_{2g + 2}$ under the standard representation. We show that when $g \geq 2$, the only closed subgroup $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ surjecting onto $\mathfrak{S}_{2g + 2}$ is its full inverse image in $\mathrm{Sp}_{2g}(\mathbb{Z}_2)$, while all subgroups $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ surjecting onto $\mathfrak{S}_{2g + 1}$ are open and contain the level-$8$ principal congruence subgroup of $\mathrm{Sp}_{2g}(\mathbb{Z}_2)$. 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.