Profinite completions of free-by-free groups contain everything

Given an arbitrary, finitely presented, residually finite group Γ, one can construct a finitely generated, residually finite, free-by-free group M_Γ = F_∞ F₄ and an embedding M_Γ (F₄ Γ)× F₄ that induces an isomorphism of profinite completions. In particular, there is a free-by-free group whose profinite completion contains Γ as a retract.