Let G be a finite group.It is proved that any class-preserving Coleman automorphism of G is an inner automorphism whenever G belongs to one of the following two classes of groups:(1) CN-groups,i.e.,groups in which the centralizer of any element is nilpotent;(2) CIT-groups,i.e.,groups of even order in which the centralizer of any involution is a 2-group.In particular,the normalizer conjecture holds for both CN-groups and CIT-groups.Additionally,some other results are also obtained.