Let $F$ be the discrete free group of rank 2, and let $\widehat{F}$ be its profinite completion, equipped with an embedding $$i : F\hookrightarrow\widehat{F}$$ By a result of Asada, this embedding induces a continuous injection $\widehat{\text{Aut}(F)}\hookrightarrow\text{Aut}\big(\widehat{F}\big)$, whose image is definitely not a normal subgroup. (One might call elements of the image "Nielsen" automorphisms of $\widehat{F}$.)

Define $\text{IAut}\big(\widehat{F}\big) := \ker\left(\text{Aut}\big(\widehat{F}\big)\rightarrow\text{Aut}\Big(\widehat{\mathbb{Z}}^2\Big)\right)$

Is $\text{IAut}\big(\widehat{F}\big)\cap\widehat{\text{Aut}(F)}$ normal inside $\text{Aut}\big(\widehat{F}\big)$?

I'd also be interested in any "philosophical remarks" that might cause someone to think one way or the other.


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