Complemented normal is quotient-transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., complemented normal subgroup) satisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property)
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Statement

Statement with symbols

Suppose $H\leq K\leq G$ 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