No nontrivial homomorphism to quotient group is not transitive

Statement
It is possible to have a finite group $$G$$ and normal subgroups $$H \le K \le G$$ such that there is no nontrivial homomorphism from $$H$$ to $$K/H$$ and no nontrivial homomorphism from $$K$$ to $$G/K$$, but there is a nontrivial homomorphism from $$H$$ to $$G/H$$.

Related facts

 * Homomorph-containment is not transitive
 * Full invariance is transitive
 * No common composition factor with quotient group is transitive

Proof
Let $$G$$ be the direct product of the symmetric group of degree three and a cyclic group of order three. Let $$K$$ be the first direct factor (i.e., the symmetric group of degree three) and $$H$$ be the unique subgroup of order three inside $$K$$.

Then, there is no nontrivial homomorphism from $$H$$ to $$K/H$$ and none from $$K$$ to $$G/K$$, but there is a nontrivial homomorphism from $$H$$ to $$G/H$$, namely, an isomorphic mapping from $$H$$ to the second direct factor of $$G$$.