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 cellularmap G-> M . We show that in general a torsion free reduced abelian group M mayhave a proper class of non-isomorphic cellular covers. In other words, thecardinality of the kernels is unbounded. In the opposite direction we show thatif the kernel of a cellular cover of any group M has certain ``freeness''properties, then its cardinality must be bounded.