SCDIN of subset-conjugacy-closed implies SCDIN

This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., SCDIN-subgroup (?) and Subset-conjugacy-closed subgroup (?)), to another known subgroup property (i.e., SCDIN-subgroup (?))
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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., subset-conjugacy-closed subgroup) must also satisfy the second subgroup property (i.e., SCDIN-subgroup)
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Statement

Suppose $H\leq K\leq G$ 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