Rationality problem for norm one tori for small non-Galois extensions: Symmetric, alternating and linear group cases including stably rational ${\rm PSL}_2(\mathbb{F}_{2^d})$ cases with $d=1,2,3$

Let $k$ be a field, $K/k$ be a separable field extension of degree $n$ and $L/k$ be the Galois closure of $K/k$. Let $G={\rm Gal}(L/k)$ and $H={\rm Gal}(L/K)\lneq G$ with $[G:H]=n$. Then $H$ is core-free in $G$, i.e. $\bigcap_{σ\in G} σ^{-1}Hσ=\{1\}$. Let $T=R^{(1)}_{K/k}(\mathbb{G}_m)$ be the norm