No common composition factor with quotient group is quotient-transitive

Statement
Suppose $$G$$ is a group of finite composition length and $$H \le K \le G$$ 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