Let $F_2$ be the free group on two generators, and $\widehat{F_2}$ its profinite completion. Let $Out(\widehat{F_2})$ be the outer automorphism group of $\widehat{F_2}$, ie, $Out(\widehat{F_2}) = Aut(\widehat{F_2})/Inn(\widehat{F_2})$. There's a natural surjection $$Aut(\widehat{F_2})\rightarrow GL_2(\widehat{\mathbb{Z}})$$ Certainly $Inn(\widehat{F_2})$ is in the kernel, so we get the map $Out(\widehat{F_2})\rightarrow GL_2(\widehat{\mathbb{Z}})$.
Furthermore, since $Out(F_2)\cong GL_2(\widehat{\mathbb{Z}})$ is known to have the congruence subgroup property, we have an injection $\widehat{Out(F_2)}\rightarrow Out(\widehat{F_2})$.
Let $G\Gamma(N)$ for $N\ge 1$ denote the congruence subgroup of $GL_2(\mathbb{Z})$ of level $N$, then we also have an exact sequence $$1\longrightarrow\bigcap\overline{G\Gamma(N)}\longrightarrow\widehat{GL_2(\mathbb{Z})}\longrightarrow GL_2(\widehat{\mathbb{Z}})$$ where $\overline{G\Gamma(N)}$ denotes the closure inside $\widehat{GL_2(\mathbb{Z})}$. Here, $\bigcap\overline{G\Gamma(N)}$ is the congruence kernel of $GL_2(\mathbb{Z})$ and is rather big (contains as a subgroup of index 2, the congruence kernel of $SL_2(\mathbb{Z})$ which is a free profinite group of countably infinite rank)
Thus, $\bigcap\overline{G\Gamma(N)}$ is also in the kernel of $Out(\widehat{F_2})\rightarrow GL_2(\widehat{\mathbb{Z}})$.
Could this be the entire story? What's known about this?
