  In this paper, a G-shift of finite type (G-SFT) is a shift of finite type
together with a free continuous shift-commuting action by a finite group G. We
reduce the classification of G-SFTs up to equivariant flow equivalence to an
algebraic classification of a class of poset-blocked matrices over the integral
group ring of G. For a special case of two irreducible components with
G$=\mathbb Z_2$, we compute explicit complete invariants. We relate our matrix
structures to the Adler-Kitchens-Marcus group actions approach. We give
examples of G-SFT applications, including a new connection to involutions of
cellular automata.
