Class-Preserving Automorphisms of Finite Groups

We construct a family of Frobenius groups having abelian Sylow subgroups and non-inner, class-preserving automorphisms. We show that any A-group (that is, a finite solvable group with abelian Sylow subgroups) has a sub-quotient belonging to provided it has a non-inner, class-preserving automorphism. As a consequence, we obtain that for metabelian A-groups, or A-groups with elementary abelian Sylow subgroups, class-preserving automorphisms are necessarily inner automorphisms. The same is true for a finite group whose Sylow subgroups of odd order are all cyclic, and whose Sylow 2-subgroups are either cyclic, dihedral, or generalized quaternion. Some applications are given.