The rationality problem for finite subgroups of GL4(Q)

Let G be a finite subgroup of GL4(Q). The group G induces an action on Q(x1,x2,x3,x4), the rational function field of four variables over Q. Theorem. The fixed subfield is rational (i.e. purely transcendental) over Q, except for two groups which are images of faithful representations of C8 and C3⋊C8 into GL4(Q) (both fixed fields for these two exceptional cases are not rational over Q). There are precisely 227 such groups in GL4(Q) up to conjugation; the answers to the rationality problem for most of them were proved by Kitayama and Yamasaki (2009) [KY] except for four cases. We solve these four cases left unsettled by Kitayama and Yamasaki; thus the whole problem is solved completely.