Is there a name for this subgroup property?

Question

Suppose that I have a subgroup $H \le G$ such that for any $h \in H$, $[h]_G \cap H = [h]_H$. Is there a name for this property?

Here, $[h]_G$ means the $G$-conjugacy class of $h$ and $[h]_H$ is the $H$-conjugacy class of $h$.

Answer 1

If $K\leq H\leq G$, we say that “$H$ controls fusion in $K$ with respect to $G$” if whenever $k,k’\in K$ are conjugate in $G$, they are also conjugate in $H$.

Your property is that $H$ controls fusion in $H$ with respect to $G$. One may also say that “$H$ controls its own fusion in $G$.”