Abelian sections of the symmetric groups with respect to their index

We show the existence of an absolute constant $\alpha>0$ such that, for every $k \geq 3$, $G:=\mathop{\mathrm{Sym}}(k)$, and $H \leqslant G$ index at least $3$, one has $|H/[H,H]| \leq |G:H|^{\alpha/ \log |G:H|}$. This inequality is best possible symmetric groups, we conjecture that it family arbitrarily large finite groups.