Subhomomorph-containment satisfies intermediate subgroup condition

Definition
Suppose $$H \le K \le G$$ are groups such that $$H$$ is a subhomomorph-containing subgroup of $$G$$. Then, $$H$$ is also a subhomomorph-containing subgroup of $$K$$.

Related facts

 * Homomorph-containment satisfies intermediate subgroup condition