The rationality problem for finite subgroups of GL_4(Q)

Let $G$ be a finite subgroup of $GL_4(\bm{Q})$. The group $G$ induces an action on $\bm{Q}(x_1,x_2,x_3,x_4)$, the rational function field of four variables over $\bm{Q}$. Theorem. The fixed subfield $\bm{Q}(x_1,x_2,x_3,x_4)^G:=\{f\in\bm{Q}(x_1,x_2,x_3,x_4):\sigma \cdot f=f$ for any $\sigma\in G\}$ is rational (i.e.\ purely transcendental) over $\bm{Q}$, except for two groups which are images of faithful representations of $C_8$ and $C_3\rtimes C_8$ into $GL_4(\bm{Q})$ (both fixed fields for these two exceptional cases are not rational over $\bm{Q}$). There are precisely 227 such groups in $GL_4(\bm{Q})$ up to conjugation; the answers to the rationality problem for most of them were proved by Kitayama and Yamasaki \cite{KY} except for four cases. We solve these four cases left unsettled by Kitayama and Yamasaki; thus the whole problem is solved completely.