Let G be a finite group. We classify G-equivariant flow equivalence of non-trivial irreducible shifts of finite type in terms of
(i) elementary equivalence of matrices over ZG and
(ii) the conjugacy class in ZG of the group of G-weights of cycles based at a fixed vertex.
In the case G = Z/2, we have the classification for twistwise flow equivalence. We include some algebraic results and examples related to the determination of E(ZG) equivalence, which involves K1(ZG).