Let $G\subseteq S_N$ be a finite permutation group and $\mathcal{P}=\bigsqcup_{i=1}^nX_i$ a partition of $\{1,\dots,N\}$.
We can define a subgroup $G_{\mathcal{P}}\subseteq G$ by: $$G_{\mathcal{P}}:=\{\sigma\in G\,\colon\,\sigma(X_i)=X_i\}.$$
In particular, the isotropy subgroup $G_i$ is given by $G_{\mathcal{P}_i}$ where $\mathcal{P}_i:=\{i\}\sqcup(\{1,\dots,N\}\backslash \{i\})$.
Question: has such a construction $G_{\mathcal{P}}\subseteq G$ got a name in the literature? If not known, can be come up with a better name than "generalised isotropy". Perhaps something with "orbits"
I see the same construction here.
It's the stabilizer subgroup of $P$, with respect to the action of $G$ on (edit: ordered / labeled) partitions.