Transfer-closed characteristicity is transitive

Definition
Suppose $$H \le K \le G$$ are groups such that $$K$$ is a transfer-closed characteristic subgroup of $$G$$ and $$H$$ is a transfer-closed characteristic subgroup of $$K$$. Then, $$H$$ is a transfer-closed characteristic subgroup of $$G$$.

Facts used

 * 1) uses::Characteristicity is transitive
 * 2) uses::Transfer condition operator preserves transitivity

Proof using given facts
The proof follows from facts (1) and (2).