Profinite completions of free-by-free groups contain everything

Given an arbitrary, finitely presented, residually finite group $\mathrm{\Gamma }$, one can construct a finitely generated, residually finite, free-by-free group $M_{\mathrm{\Gamma }}=F_{\mathrm{\infty }}⋊F_4$ and an embedding $M_{\mathrm{\Gamma }}\hookrightarrow (F_4\ast \mathrm{\Gamma })\times F_4$ that induces an isomorphism of profinite completions. In particular, there is a free-by-free group whose profinite completion contains $\widehat{\mathrm{\Gamma }}$ as a retract.