An epimorphism $\phi :\,G\,\to \,H$ of groups, where $G$ has rank $n$, is called coessential if every (ordered) generating $n$-tuple of $H$ can be lifted along $\phi $ to a generating $n$-tuple for $G$. We discuss this property in the context of the category of groups, and establish a criterion for such a group $G$ to have the property that its abelianization epimorphism $G\,\to \,{G}/{[G,G]}\;$, where $[G,\,G]$ is the commutator subgroup, is coessential. We give an example of a family of 2-generator groups whose abelianization epimorphism is not coessential. This family also provides counterexamples to the generalized Andrews–Curtis conjecture.