  In this paper, we show that all Coleman automorphisms of a finite group with
self-central minimal non-trivial characteristic subgroup are inner; therefore
the normalizer property holds for these groups. Using our methods we show that
the holomorph and wreath product of finite simple groups, among others, have no
non-inner Coleman automorphisms. As a further application of our theorems, we
provide partial answers to questions raised by M. Hertweck and W. Kimmerle.
Furthermore, we characterize the Coleman automorphisms of extensions of a
finite nilpotent group by a cyclic $p$-group. Lastly, we note that
class-preserving automorphisms of 2-power order of some nilpotent-by-nilpotent
groups are inner, extending a result by J. Hai and J. Ge.
