Direct factor is quotient-transitive

Statement
Suppose $$H \le K \le G$$ are groups such that $$H$$ is a direct factor of $$G$$ and $$K/H$$ is a direct factor of the quotient group $$G/H$$. Then $$K$$ is a direct factor of $$G$$.

Related facts

 * Complemented normal is quotient-transitive
 * Central factor is not quotient-transitive
 * Central factor over direct factor implies central factor