Direct factor satisfies intermediate subgroup condition

Statement
Suppose $$H \le K \le G$$ are groups such that $$H$$ is a direct factor of $$G$$. Then, $$H$$ is also a direct factor of $$K$$.

Related facts

 * Permutably complemented satisfies intermediate subgroup condition
 * Complemented normal satisfies intermediate subgroup condition
 * Retract satisfies intermediate subgroup condition