No common composition factor with quotient group is quotient-transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., normal subgroup having no common composition factor with its quotient group) satisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property)
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Statement

Suppose $G$ is a group of finite composition length and $H\leq K\leq G$ are normal subgroups such that $H$ has no common composition factors with $G/H$ and $K/H$ has no common composition factors with $G/K$. Then, $K$ has no common composition factors with $G/K$.

Related facts

• Normal Hall is quotient-transitive
• No common composition factor with quotient group is transitive