Class-preserving automorphisms and the normalizer property for Blackburn groups

For a group G, let U be the group of units of the integral group ring ZG. The group G is said to have the normalizer property if N U G ZUG. It is shown that Blackburn groups have the normalizer property. These are the groups which have non-normal finite subgroups, with the intersection of all of them being non-trivial. Groups G for which classpreserving automorphisms are inner automorphisms, Out c G 1, have the normalizer property. Recently, Herman and Li have shown that Out c G 1 for a finite Blackburn group G. We show that Out c G 1 for the members G of certain classes of metabelian groups, from which the Herman-Li result follows.