CDIN of conjugacy-closed implies CDIN

This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., CDIN-subgroup (?) and Conjugacy-closed subgroup (?)), to another known subgroup property
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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., conjugacy-closed subgroup) must also satisfy the second subgroup property (i.e., right-transitively CDIN-subgroup)
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Statement

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