In summer 2001 we used Jon Carlson's computer computation for group cohomology
of 2-groups of order up to 64 to study the problem. We
discovered a group G of order 64 and an 'almost inner' but non inner
automorphism of this group inducing the identity on the mod 2 group
cohomology. In the meantime Kasper Andersen communcated to us that he
studied the list more systematically and discovered 17 such examples.