Finite direct power-closed characteristic is transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., finite direct power-closed characteristic subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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Statement

Statement with symbols

Suppose $H\leq K\leq G$ are groups such that $H$ is a finite direct power-closed characteristic subgroup of $K$ and $K$ is a finite direct power-closed characteristic subgroup of $G$. Then, $H$ is a finite direct power-closed characteristic subgroup of $G$.