Complemented normal is quotient-transitive

Statement with symbols
Suppose $$H \le K \le G$$ are groups such that $$H$$ is a complemented normal subgroup of $$G$$ and $$K/H$$ is a complemented normal subgroup of $$G$$. Then $$K$$ is a complemented normal subgroup of $$G$$.

Related facts

 * Direct factor is quotient-transitive
 * Central factor over direct factor implies central factor