Let $\widehat{F_2}$ be the pro-$\ell$ completion of the free group of rank 2, where $\ell$ is some prime.

Every outer automorphism of $F_2$ induces an outer automorphism of $\widehat{F_2}$, hence an injection $Out(F_2)\rightarrow Out(\widehat{F_2})$.

Of course in $Out(F_2)$ there are plenty of automorphisms of finite order. My question is: Does every automorphism of $Out(\widehat{F_2})$ of finite order lie in the closure $\overline{Out(F_2)}$ inside $Out(\widehat{F_2})$?


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