Subset-conjugacy-determined subgroup

Definition
Suppose $$H \le K \le G$$ are groups. We say that $$H$$ is subset-conjugacy-determined in $$K$$, or that fusion of subsets of $$H$$ in $$G$$ is contained in $$K$$, if whenever $$A,B \subseteq H$$ and $$g \in G$$ is such that $$gAg^{-1} = B$$, there exists $$k \in K$$ such that $$kak^{-1} = gag^{-1}$$ for all $$a \in A$$.

If $$H$$ is subset-conjugacy-determined in itself relative to $$G$$, we say that $$H$$ is a subset-conjugacy-closed subgroup.

Weaker properties

 * Stronger than::Conjugacy-determined subgroup
 * Stronger than::Normal subset-conjugacy-determined subgroup
 * Stronger than::Weak normal subset-conjugacy-determined subgroup

Related subgroup properties

 * Subset-conjugacy-closed subgroup: A subgroup that is subset-conjugacy-determined in itself relative to the whole group.
 * SCDIN-subgroup: A subgroup that is subset-conjugacy-determined in its normalizer, relative to the whole group.

Facts

 * Center of pronormal subgroup is subset-conjugacy-determined in normalizer