No nontrivial homomorphism to quotient group is not transitive

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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 \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

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.