Homomorph-containment is quotient-transitive

Statement with symbols
Suppose $$G$$ is a group, and $$H \le K \le G$$ are subgroups of $$G$$ such that $$H$$ is a homomorph-containing subgroup of $$G$$ and $$K/H$$ is a homomorph-containing subgroup of $$G/H$$. Then, $$K$$ is a homomorph-containing subgroup of $$G$$.

Related facts about homomorph-containing subgroups

 * Homomorph-containment is not transitive
 * Homomorph-containment satisfies intermediate subgroup condition

Related facts about quotient-transitivity

 * Full invariance is quotient-transitive
 * Strict characteristicity is quotient-transitive
 * Characteristicity is quotient-transitive
 * Normality is quotient-transitive