On kernels of cellular covers

In the present paper we continue to examine cellular covers of groups, focusing on the cardinality and the structure of the kernel K of the cellular map G -> M. We show that in general a torsion free reduced abelian group M may have a proper class of non- isomorphic cellular covers. In other words, the cardinality of the kernels is unbounded. In the opposite direction we show that if the kernel of a cellular cover of any group M has certain "freeness" properties, then its cardinality is bounded by vertical bar M vertical bar.