I've stumbled upon a construction which seems to be very much classical, and yet I found nothing definite about it so far in available sources. Let $\Lambda$ be a normal subgroup of the automorphism group $\mathrm{Aut}(G)$ of a group $G.$ Set $$ \mathrm{Res}(\Lambda)=\mathrm{nc}( g^{-1} \lambda(g) : g \in G, \lambda \in \Lambda) $$ (the normal closure in the right-hand side; 'Res' stands for the 'residue', like in a similar construction in the theory of linear groups). Then the quotient group $\mathrm{Aut}(G)/\Lambda$ acts on the quotient group $G/\mathrm{Res}(\Lambda).$

I would be grateful for a reference to a text where some basic facts on the above construction can be found. There is a couple of questions I would like to have any information about:

1) Under what conditions $\mathrm{Aut}(G)/\Lambda$ is the full automorphism group of the group $G/\mathrm{Res}(\Lambda)?$ (It may easily happen that $\mathrm{Res}(\Lambda_1)=\mathrm{Res}(\Lambda_2),$ with obvious consequences);

2) What happens if one switches from $\Lambda$ to an isomorphic normal subgroup $\Lambda' \cong \Lambda$ of $\mathrm{Aut}(G)?$


I would venture that this is somewhat studied when $\Lambda$ is the subgroup of inner automorphisms, and so $Aut(G)/\Lambda = Out(G)$, the outer automorphism group, and $G / Res(\Lambda) = Ab(G)$, the abelianization of $G$.

Let me address question 1) rather narrowly with two examples.

Example: when $G = \pi_1(S)$ for $S$ a closed surface of genus $g \ge 2$, $Out(G)$ is the mapping class group of $S$, and $Ab(G) = Z^{2g}$. The action of $Out(G)$ is not all of $Aut(Z^{2g}) = GL(2g,Z)$ but is instead the symplectic group. What happens here is that there is extra structure on $Ab(G)$ that is preserved by the action, namely the intersection product of cycles on $S$.

For another example when $G = F_n$, for $F_n$ a free group of finite rank $n$, where $Ab(F_n)=Z^n$, the action of $Out(F_n)$ is indeed all of $Aut(Z^n) = GL(n,Z)$.

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