Subhomomorph-containing implies right-transitively homomorph-containing

Statement with symbols
Suppose $$H \le K \le G$$ are groups. If $$H$$ is a homomorph-containing subgroup of $$K$$ and $$K$$ is a subhomomorph-containing subgroup of $$G$$.

Related facts

 * Homomorph-containment is not transitive
 * Subhomomorph-containment is transitive