I am studying group actions and trying to organize some concepts with each other. Below, I write my classification of 3 cases. I am wondering whether this organization can get more details.
Suppose $G$ is a group, $N_G(H)$ is the normalizer of $H$, and $C_G(H)$ is the centralizer of $H$. Also, suppose $\tau_g$ represents an action on set $H$ such that $\tau_g(h)=ghg^{-1}$.
I have organized actions of some quotient groups on the group of all permutations of $H$ ($A(H)$) as follows:
I: $\begin{cases}H \lhd G \\ H: \text{abelian}\end{cases}$
We can define homomrphism $\tau:\ G/H \rightarrow A(H)$, where $\tau(gH)=\tau_g$. With the fact that $H$ is normal, $\tau_g \in A(H)$. With the fact that $H$ is abelian, $\tau$ is well-defined.
II: $H \lhd G$
We can define homomrphism $\tau:\ G/C_G(H) \rightarrow A(H)$, where $\tau(gC)=\tau_g$. With the fact that $H$ is normal, $\tau_g \in A(H)$. However, since $H$ is not necessarily abelian, we need to change from $G/H$ to $G/C_G(H)$ so that $\tau$ is well-defined.
III: $H <G$
We can define homomrphism $\tau:\ N_G(H)/C_G(H) \rightarrow A(H)$, where $\tau(nC)=\tau_n$. Since $H$ is not necessarily normal, we need to change from $G/C_G(H)$ to $N_G(H)/C_G(H)$ so that $\tau_n \in A(H)$. Also, $\tau$ is monomorphism in this case.
Now, can we extend these cases to the fourth case when $H$ is just a subset of $G$? Do you have something to add to this classification?