This is a beautiful example. Justin Moore and myself show that $F_\tau$ does not embed into $F$ in arxiv.org/abs/2103.14911

Let $X$ be a finite symmetric generating set for $G$ and let $\phi:X^* \to G$ be the canonical homomorphism. Let $w$ be in $(f)\phi^{-1}$. Now, $u$ is in $(M)\phi^{-1}$ exactly if $u^{-1}wu$ is in $H$ which we can check because $H$ is RE.