No nontrivial homomorphism to quotient group is not transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., normal subgroup having no nontrivial homomorphism to its quotient group) not satisfying a subgroup metaproperty (i.e., transitive subgroup property).
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Statement

It is possible to have a finite group $G$ and normal subgroups $H\leq K\leq G$ 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 $G$.