Metabolic modules of finite group algebras over finite fields of characteristic two

Let G be a finite group and let F be a finite field of characteristic 2. We introduce F-special subgroups and F-special elements of G. In the case where F contains a pth primitive root of unity for each odd prime p dividing the order of G (e.g. it is the case once F is a splitting field for all subgroups of G), the F-special elements of G coincide with real elements of odd order. We prove that a symmetric FG-module V   is metabolic if and only if the restriction VDVD of V to every F-special subgroup D of G is metabolic, and also, if and only if the characteristic polynomial on V defined by every F-special element of G is a square of a polynomial over F. Some immediate applications to characters, self-dual codes and Witt groups are given.