Let groups $G$ and $H$ act on a set $X$. Then $$ \{g\in G\mid \forall x\in X, h\in H: ghx = hgx\} \le G $$ is a subgroup of $G$.
Does this subgroup have a name, or any particular significance?
This is just a question out of curiousity.
Context: View the action by $H$ as a functor $F:\mathbf{B}H\to \mathbf{Set}$ from the one-object group category $\mathbf{B}H$. Then my subgroup is the set of elements $g\in G$ such that $(x\mapsto gx)$ is a natural transformation $F\Rightarrow F$.
This came up randomly in a calculation, but may allude to there being some importance to this subgroup.
It's the intersection of $G$ and $\text{Aut}_H(X)$. That's also what your natural transformation computation says.