Let $F$ be a finitely generated free group and let $A$ be a finite index subgroup of $F$. Does there exist a subgroup $B\subset A$ such that $F/B$ is (elementary) amenable and torsion-free?

A group $G$ which is amenable and torsion-free has (at least conjecturally) the following nice properties: $\Bbb{Z}[G]$ has conjecturally no zero divisors (this is known if $G$ is say locally indicable or left-orderable) and $\Bbb{Z}[G]$ embeds in its Ore localization.

I have a certain application for the above question in mind, where using an Ore localization plays a role. But I am also curious since my intuition fails me on that question. Note that if $F/A$ is solvable, then one could just take $F$ to be an iterated commutator subgroup. But if $F/A$ is non-solvable, then I don't know what to do.