CDIN of conjugacy-closed implies CDIN

Statement with symbols
Suppose $$H \le K \le G$$ are groups such that $$H$$ is a CDIN-subgroup of $$K$$ and $$K$$ is a conjugacy-closed subgroup of $$G$$. Then $$H$$ is a CDIN-subgroup of $$G$$.

Related facts

 * SCDIN of subset-conjugacy-closed implies SCDIN