SCDIN of subset-conjugacy-closed implies SCDIN

Statement
Suppose $$H \le K \le G$$ are groups such that $$H$$ is a SCDIN-subgroup of $$K$$ and $$K$$ is a subset-conjugacy-closed subgroup of $$G$$. Then, $$H$$ is a SCDIN-subgroup of $$G$$.

Related facts

 * CDIN of conjugacy-closed implies CDIN
 * Characteristic of SCDIN implies SCDIN
 * Normalizer-relatively normal of SCDIN implies SCDIN
 * Characteristic of CDIN implies CDIN
 * Normalizer-relatively normal of CDIN implies CDIN