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 $$\mathfrak {S}_{2g + 1} {F}_2)$$ of $$S_{2g 1}$$ or 2} 2}$$ under standard representation. show that when $$g \ge 2$$ , only subgroup surjecting onto is its full inverse in $$\mathrm while all are open and contain level-8 principal congruence . As an immediate application, we able to strengthen a result Zarhin on 2-adic Galois representations associated hyperelliptic curves. also prove elementary corollary concerning even-degree polynomials group.