Characteristic of CDIN implies CDIN

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

Related facts

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