Let G denote a nite group. In this paper, by a G shift of nite type ( G-SFT) we will mean a shift of nite type (SFT) together with a continuous G-action which commutes with the shift, where in addition the action is free and the SFT is irreducible and is non-trivial (contains more than one orbit). We will classify these systems up to G-5ow equivalence. This equivalence relation can be described in terms of G-SFTs, skew products or suspension 5ows (x 2). For example, two G-SFTs are G-5ow equivalent if and only if there exists an orientation-preserving homeomorphism between their mapping tori which commutes with the induced G actions. A G-SFT can be presented by a nite square matrix A over ZþG, the positive cone of the integral group ring ZG (W. Parry, personal communication 2001). Let ðIAÞ1 denote the NN matrix whose upper left corner is IA and which otherwise equals the innite identity matrix. Let E ðZGÞ be the group of NN matrices generated by basic elementary matrices (those which di;er from I in at most one entry, which must be o;-diagonal) over ZG. Let WðAÞ denote the weight class of A (Denition 4.1): the conjugacy class in G of the group of weights of loops based at a xed vertex. We show that the weight class is an invariant of G-5ow equivalence. When WðAÞ¼WðBÞ¼G, we will show that G-SFTs presented by matrices A and B are G-5ow equivalent if and only if there are matrices U and V in EðZGÞ such that UðIAÞ1V ¼ðIBÞ1 (Theorem 6.1). The complete classication up to G-5ow equivalence, which allows the possibility WðAÞ ( G, has a more complicated statement (Theorem 6.4). In the case that G is trivial, our classication reduces to the familiar classication of Franks ( 14) by cokernel group and determinant. When G is non-trivial, the classication up to E ðZGÞ is much more diBcult and interesting, and remains an open problem. We consider these algebraic issues inxx 8 and 9. In x 8, we give the modest requisite K-theory terminology and background, and for the case G¼ Z=2 we give a constructive partial result (Theorem 8.1) and some very concrete illustrative examples (Examples 8.6 and 8.7) which indicate how the ZG-equivalence problem becomes more diBcult when G is non-trivial (that is, ZG6 Z). In x 9, we consider EðZGÞ-equivalence of injective matrices. In this case, GLðZGÞ-equivalence amounts to isomorphism of cokernel modules, and the renement toE ðZGÞ-equivalence is classied by K 1ðZGÞ=H for an associated subgroup H ofS K 1ðZGÞ. As one consequence, if G is abelian and detðIAÞ is not a zerodiviso r in ZG, then detðIAÞ determines the G-5ow equivalence class up