Reduction of UNil for finite groups with normal abelian Sylow 2-subgroup

Let FF be a finite group with a Sylow 2-subgroup SS that is normal and abelian. Using hyperelementary induction and cartesian squares, we prove that Cappell’s unitary nilpotent groups UNil∗(Z[F];Z[F],Z[F]) have an induced isomorphism to the quotient of UNil∗(Z[S];Z[S],Z[S]) by the action of the group F/SF/S. In particular, any finite group FF of odd order has the same UNil-groups as the trivial group. The broader scope is the study of the LL-theory of virtually cyclic groups, based on the Farrell–Jones isomorphism conjecture. We obtain partial information on these UNil when SS is a finite abelian 2-group and when SS is a special 2-group.