Abelian-quotient not implies cocentral

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., abelian-quotient subgroup) need not satisfy the second subgroup property (i.e., cocentral subgroup)
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Statement

It is possible to have a group $G$ and a normal subgroup $H$ such that $G/H$ is an abelian group, so $H$ is an abelian-quotient subgroup, but $HZ(G)$ is not equal to $G$, i.e., $H$ is not a cocentral subgroup of $G$.