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 $k\ge 3$, $G:=(k)$, and for every $H⩽G$ of index at least $3$, one has $\mathrm{\mid }H\mathrm{/}[H,H]\mathrm{\mid }\le \mathrm{\mid }G:H{\mathrm{\mid }}^{\alpha \mathrm{/}\log \log \mathrm{\mid }G:H\mathrm{\mid }}$. This inequality is the best possible for the symmetric groups, and we conjecture that it is the best possible for every family of arbitrarily large finite groups.