  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 must be bounded.
