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 $\mathrm{Aut}(G)?$