The Congruence Subgroup Problem for low rank Free and Free Metabelian groups

The congruence subgroup problem for a finitely generated group $\mathrm{\Gamma }$ asks whether $\widehat{Aut\left(\mathrm{\Gamma }\right)}\to Aut(\widehat{\mathrm{\Gamma }})$ is injective, or more generally, what is its kernel $C\left(\mathrm{\Gamma }\right)$? Here $\widehat{X}$ denotes the profinite completion of $X$. In this paper we first give two new short proofs of two known results (for $\mathrm{\Gamma }=F_2$ and ${\mathrm{\Phi }}_2$) and a new result for $\mathrm{\Gamma }={\mathrm{\Phi }}_3$:

1. $C\left(F_2\right)=\left\{e\right\}$ when $F_2$ is the free group on two generators.
2. $C\left({\mathrm{\Phi }}_2\right)={\widehat{F}}_{\omega }$ when ${\mathrm{\Phi }}_n$ is the free metabelian group on $n$ generators, and ${\widehat{F}}_{\omega }$ is the free profinite group on ${\mathrm{\aleph }}_0$ generators.
3. $C\left({\mathrm{\Phi }}_3\right)$ contains ${\widehat{F}}_{\omega }$. Results 2. and 3. should be contrasted with an upcoming result of the first author showing that $C\left({\mathrm{\Phi }}_n\right)$ is abelian for $n\ge 4$.