Rationality of fields of invariants of some four-dimensional linear groups, and an equivariant construction related to the Segre cubic

Let $G\subset SL(4)$ be a finite primitive linear group. We prove that if $G$ contains a normal subgroup of order 32 then the quotient variety $\mathbf P^3/G$ is birationally isomorphic to $X/G$, where $X$ is the Segre cubic. We also prove the rationality of $\mathbf P^3/G$ for a large class of such groups (in particular, solvable groups).