Automorphisms Fixing Every Normal Subgroup of a Nilpotent-by-Abelian Group

Among other things, we prove that the group of automorphisms fixing every normal subgroup of a (nilpotent of class c )-by-abelian group is (nilpotent of class ≤ c )-by-metabelian. In particular, the group of automorphisms fixing every normal subgroup of a metabelian group is soluble of derived length at most 3 . An example shows that this bound cannot be improved.