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).
Boyle, Mike and Sullivan, Michael C. "Equivariant Flow Equivalence of Shifts of Finite Type by Matrix Equivalence over Group Rings." (Jan 2005).