Let $F$ be a free finitely generated group, $H \leq F$ of infinite index. Let $c : F \rightarrow \hat{F}$ be the embedding in the profinite completion. Denote by $\tilde{F}, \tilde{H}$ the closure of $c(F), c(H)$ respectively. Is it possible that $[\tilde{F} : \tilde{H}] < \infty$?

Does it change anything if $H$ is finitely generated?

infinitefinitely generated group with no finite quotients...' – HJRW Aug 1 at 16:11helpfullypoints out, I should in fact have written 'take $Q$ to be an infinite finitely generated group with nonon-trivialfinite quotients...' – HJRW Aug 2 at 2:14