  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.
