We study some families of finite groups having inner class-preserving
automorphisms. In particular, let G be a finite group and S be a
semidihedral Sylow 2-subgroup. Then, in both cases when either Sym(4) is not
a homomorphic image of G and $$Z(S) < Z(G)$$
Z
(
S
)
<
Z
(
G
)
or G is nilpotent-by-nilpotent, we
have that all the Coleman automorphisms of G are inner. As a consequence, these
groups satisfy the normalizer problem.