Normalizer-relatively normal of CDIN implies CDIN

Statement with symbols
Suppose $$H \le K \le G$$ are groups such that $$H$$ is a normalizer-relatively normal subgroup of $$K$$ relative to $$G$$, and $$K$$ is a CDIN-subgroup of $$G$$. Then, $$H$$ is a CDIN-subgroup of $$G$$.

Related facts

 * Characteristic of CDIN implies CDIN
 * Normalizer-relatively normal of SCDIN implies SCDIN
 * Characteristic of SCDIN implies SCDIN
 * Characteristic central factor of WNSCDIN implies WNSCDIN
 * Characteristic central factor of MWNSCDIN implies MWNSCDIN