T-groups,polycyclic groups,and finite quotients

A group is called a T-group if all of its subnormal subgroups are normal. In this note we consider the following question: Assume that G is a polycyclic group. What can be said about G if all finite epimorphic images H of G satisfy one of the following conditions: (i) H is a T-group,(ii) ({H/Phi (H)}) is a T-group,(iii) H/Z _*(H) is a T-group. We will see that the prime 2 will play a particular role in (ii) and (iii), see Theorems C and D.