Coleman automorphisms of finite groups with semidihedral Sylow 2-subgroups

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 .