Invariants of wreath products and subgroups of S₆

Let $G$ be a subgroup of $S_6$, the symmetric group of degree 6. For any field $k$, $G$ acts naturally on the rational function field $k(x_1,...,x_6)$ via $k$-automorphisms defined by $\sigma\cdot x_i=x_{\sigma(i)}$ for any $\sigma\in G$, any $1\le i\le 6$. Theorem. The fixed field $k(x_1,...,x_6)^G$ is rational (=purely transcendental) over $k$, except possibly when $G$ is isomorphic to $PSL_2(\bm{F}_5)$, $PGL_2(\bm{F}_5)$ or $A_6$. When $G$ is isomorphic to $PSL_2(\bm{F}_5)$ or $PGL_2(\bm{F}_5)$, then $\bm{C}(x_1,...,x_6)^G$ is $\bm{C}$-rational and $k(x_1,...,x_6)^G$ is stably $k$-rational for any field $k$. The invariant theory of wreath products will be investigated also.