A G-shift of finite type (G-SFT) is a shift of finite type which commutes with the continuous action of a finite group G. We classify irreducible G-SFTs up to right closing almost conjugacy, answering a question of Bill Parry.

Then, we derive a computable set of necessary and sufficient conditions for the existence of a homomorphism from one shift of finite type to another. We consider an equivalence relation on subshifts, called weak equivalence, which was introduced and studied by Beal and Perrin. We classify arbitrary shifts of finite type up to weak equivalence.