Direct factor is quotient-transitive

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

Suppose $H\leq K\leq G$ 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