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