Noether's problem for some subgroups of $S_{14}$: the modular case

Let $G$ be a subgroup of $S_n$, the symmetric group of degree $n$. For any field $k$, $G$ acts naturally on the rational function field $k(x_1,\cdots ,x_n)$ via $k$-automorphisms defined by $\sigma\cdot x_{i}:=x_{\sigma\cdot i}$forany$\sigma\in G$and$1\leq i\leq n$. In this article, we will show that if $G$ is a solvable transitive subgroup of $S_{14}$ and $\text{char}(k)=7$, then the fixed subfield $k(x_1,\cdots ,x_{14}{)}^G$ is rational (i.e., purely transcendental) over $k$. In proving the above theorem, we rely on the Kuniyoshi-Gasch"utz Theorem or some ideas in its proof.