Abstract: This is a continuation of an earlier preprint 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 state two conjectures. A direct application of one of the conjectures to the virtual Betti number conjecture is mentioned. The other conjecture says that most nonpositively curved Riemannian manifold have fundamental groups which are not eventually perfect. For completeness, here we reproduce parts of the previous paper.