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Among other things, we prove that the group of automorphisms fixing everynormal subgroup of a nilpotent-by-abelian group is nilpotent-by-metabelian. Inparticular, the group of automorphisms fixing every normal subgroup of ametabelian group is soluble of derived length at most 3. An example shows thatthis bound cannot be improved.

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