Weak marginality is quotient-transitive

Statement
Suppose $$H \le K \le G$$ are groups such that $$H$$ is a weakly marginal subgroup of $$G$$ and $$K/H$$ is a weakly marginal subgroup of $$G/H$$. Then, $$K$$ is a weakly marginal subgroup of $$G$$.

Proof
The proof idea is to substitute one word-letter pair inside the other.