  This is a continuation of an earlier preprint (math.GT/0209121) under the
same title. These papers grew out of an attempt to find a suitable finite
sheeted covering of an aspherical 3-manifold so that the cover either has
infinite or trivial first homology group. With this motivation we defined a new
class of groups. These groups are in some sense eventually perfect. Here we
prove results giving several classes of examples of groups which do (not)
belong to this class. Also we prove some basic results on these groups and
state two conjectures. A direct application of one of the conjectures to the
virtual Betti number conjecture is mentioned. For completeness, here we
reproduce parts of math.GT/0209121.
