Conjugacy-determined subgroup

Definition
Suppose $$H \le K \le G$$ are groups. We say that $$H$$ is conjugacy-determined in $$K$$ relative to $$G$$, or that $$K$$ contains fusion of elements of $$H$$ in $$G$$, if two elements of $$H$$ are conjugate in $$K$$ if and only if they are conjugate in $$G$$.

If $$H$$ is conjugacy-determined in itself relative to $$G$$, $$H$$ is termed a conjugacy-closed subgroup of $$G$$.

Stronger properties

 * Weaker than::Subset-conjugacy-determined subgroup