Subgroups forced by quotients of symmetric groups

For finite groups $H_1,H_2$, write $H_1\implies H_2$ when every finite group having a quotient isomorphic to $H_1$ also has a subgroup isomorphic to $H_2$. The symmetric-group quotient-to-subgroup conjecture. For every positive integer $n$, there exists a positive integer $N$ such that

$$S_N\implies S_n.$$

This conjecture asks whether sufficiently large symmetric quotients force smaller symmetric subgroups in arbitrary finite extensions. The paper notes that no examples were known with a non-abelian group on the subgroup side and presents this as a conjectural counterpoint; it remains open.