So awhile back I asked this question here on stack exchange:
Normal subgroup $H$ of $G$ with same orbits of action on $X$.
At the time I wasn't quite sure what I was really wanting to know about that situation. After much searching around and occasionally revisiting the problem that sparked that question, I eventually figured what the name for the thing I was noticing that peaked my interest is: It's the notion of a block system for a group action.
With that in mind, I am now prepared to ask a much more specific question:
Suppose we have a group action of a group $H$ on a set $X$, with proper nontrivial normal subgroups $M$ and $N$. What is the significance of the block systems induced by restriction of the group action to the normal subgroups $M$ and $N$ being the same (that is, the induced actions of $M$ and of $N$ on $X$ have the same block system on $X$)?
If there isn't much that can be said about this in general, then my specific case of interest is when the set is a given group finite $G$, and the acting group is its automorphism group, $\operatorname{Aut}(G)$. The normal subgroups I am specifically interested in are the inner automorphism group $\operatorname{Inn}(G)$ and the group of (conjugacy) class-preserving automorphisms, which I will denote as $\Lambda_{id}(G)$.