There are four Bowen–Franks groups associated to each shift of finite type. For an irreducible shift of finite type, we show that a 4-tuple of automorphisms corresponding to the four Bowen–Franks groups can be induced simultaneously by a specific path of flow equivalence from the shift to itself, if and only if it is F-compatible. The F-compatibleness defined in this paper describes completely the intrinsic relations among the four automorphisms induced by a flow equivalence. This result is one of the key ingredients in classifying reducible shifts of finite type up to flow equivalence. In the mean time, it also discloses a new and sharp difference between the invariants of flow equivalence for an irreducible shift of finite type, and the invariants of stable isomorphism for the associated simple Cuntz–Krieger algebra.