Let $F$ be the free group of rank $\aleph_0$, and let $L \leq F$ be a finite index subgroup. Denote by $M = L' = [L,L]$ the commutator subgroup of $L$. Is it possible that there exists a finite subset $S \subseteq F$ for which $\langle M,S \rangle = F$  ?

Note that if $L = F$ then the answer is clearly negative.

Let $F$ be the free group of rank $\aleph_0$, let $L \leq F$ be a finite index subgroup, and let $M$ be a proper characteristic subgroup of $L$. Is it possible that there exists a finite subset $S \subseteq F$ for which we have $\langle M,S \rangle = F$?