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.