Abelian-quotient not implies cocentral

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 $$G$$.