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Let $F$ be the free profinite group on two generators. Let $\text{IA}(F) := \ker\left(\text{Aut}(F)\rightarrow GL_2(\widehat{\mathbb{Z}})\right)$, the group of "IA automorphisms" of $F$. (I'm also happy to consider $\text{Out}(F)$ rather than $\text{Aut}(F)$, though there shouldn't be much difference.)

Here, the image of an automorphism of $F$ in $GL_2(\widehat{\mathbb{Z}})$ tells you how it acts on abelian quotients. It seems to me there should morally be another "part" of $\text{Aut}(F)$ which describes how automorphisms of $F$ act on nonabelian quotients of $F$, and moreover that for some notion of an "extremely nonabelian" quotient, the abelian and nonabelian "parts" of $\text{Aut}(F)$ should be "orthogonal".

To what extent is this correct/reasonable?

I also don't know what "extremely nonabelian" should mean, but we'll let ENA denote such a property. A reasonable candidate for ENA is "perfect", or "all simple composition factors are finite simple nonabelian groups". Let $\Delta$ be this second property, then $\Delta$ has the benefit that the set of $\Delta$-groups are a formation of finite groups (ie, closed under quotients and subdirect products), and so one may speak of the pro-$\Delta$ completion of a group, and so on. It's unclear to me if 2-generated perfect groups also form a formation.

My first specific question is this: Let $F\twoheadrightarrow G$ be an ENA quotient. Let $K_G$ be the intersection of all $G$-defining subgroups of $F$, then $K_G$ is characteristic in $F$ and we have a surjection $$p_G : \text{Aut}(F)\longrightarrow\text{Aut}(F/K_G)$$ For which values of ENA is the restriction $p_G|_{\text{IA}(F)}$ also surjective?

My second question is this: What is known about the maximal pro-$\Delta$ quotient of $F$? (or similar objects. I don't even know what to google!) If $F^\Delta$ denotes the maximal pro-$\Delta$ quotient of $F$, then we get a surjection $F\twoheadrightarrow F^\Delta$ with characteristic kernel, and again there is a surjective map $$p_\Delta : \text{Aut}(F)\longrightarrow\text{Aut}(F^\Delta)$$ Is $p_\Delta|_{\text{IA}(F)}$ surjective? What if we replace $\Delta$ with other possible values of ENA?

My last question is: Is it reasonable to expect that the map $\text{Aut}(F)\longrightarrow \text{Aut}(F^\Delta)\times GL_2(\widehat{\mathbb{Z}})$ is injective? Surjective? What if we replace $\Delta$ with other possible values of ENA?

I'd very much appreciate any references or results related to the questions.

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$\newcommand{\Zhat}{\widehat{\mathbb{Z}}}$

Here are some partial answers, though I would welcome any additional input.

There is a natural map $$p : F\rightarrow F^\Delta\times\Zhat^2$$ The kernel of $F\rightarrow\Zhat^2$ is just $[F,F]$, and since all finite quotients of $F^\Delta$ are perfect, $[F,F]$ surjects onto $F^\Delta$, and thus the map $p$ above is surjective. Since the all finite simple quotients of $F$ must factor through $F^\Delta\times\Zhat^2$, the kernel of $p$ is contained in the intersection of all maximal normal subgroups of $F$, otherwise known as the "Jacobson radical" of $F$ (also known as the Baer radical, or small radical). This is similar to the Frattini subgroup $\Phi(F)$, but in this case it cannot equal to $\Phi(F)$, since by Corollary 8.7.5 in Ribes/Zalesskii, $\Phi(F) = 1$ (thanks to Benjamin Steinberg for the reference), and by Proposition 8.7.7, $F$ is not a direct product.

Nonetheless, we get maps $\newcommand{\Aut}{\text{Aut}}$ $$\Aut(F)\twoheadrightarrow\Aut(F^\Delta\times\Zhat^2)\cong \Aut(F^\Delta)\times\Aut(\Zhat^2)$$ where the isomorphism is due to the fact that the projections of $F^\Delta\times\Zhat^2$ onto each direct factor is a characteristic quotient.

Thus, if $G$ is a 2-generated pro-$\Delta$ group, then any surjection $F\rightarrow G$ must factor through $F^\Delta$. This implies a positive answer to the first question if ENA = $\Delta$ and $G$ is pro-$\Delta$. Actually, by similar arguments, as long as $F/K_G$ is perfect, the answer to the first question is also positive.

This also gives a positive answer to the second question.

For the third question, the above shows that the map is definitely surjective, though not injective - for example, inner automorphisms by elements in the Jacobson radical of $F$ lie in the kernel.

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