I have a somewhat broad question related to group actions and their restriction to a normal subgroup. If we have a group action $\sigma : G \times X \rightarrow X$ with orbits $G_x$, and a normal subgroup, $H$ of $G$, such that the restriction of the action $\sigma$ to $H$, $\sigma |_H : H \times X \rightarrow X$ has the same orbits, $H_x=G_x$. What can we say about $H$ as a subgroup of $G$, and about it's relationship to $G$'s action on $X$?
Edit: The particular case I am interested in involves the action of $\operatorname{Aut}(G)$ on $G$, specifically I am interested in two subgroups, which are both characteristic (and hence normal) in $\operatorname{Aut}(G)$. One is the inner automorphism group, $\operatorname{Inn}(G)$, the other is the group of class-preserving automorphisms (those automorphisms which map conjugacy classes of $G$ to themselves), which I will denote as $\Lambda_{id}(G)$. I have already established that $\operatorname{Inn}(G) \trianglelefteq \Lambda_{id}(G)$ (follows from their normality in $\operatorname{Aut}(G)$, and that every inner automorphism is class-preserving). The interesting thing about the restricted actions, $\rho : \operatorname{Inn}(G) \times G \rightarrow G$ and $\pi : \Lambda_{id}(G) \times G \rightarrow G$, is that they both have the conjugacy classes of $G$ as orbits.