Rationality problem for norm one tori for $A_5$ and ${\mathrm{P}\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_8)$ extensions

We give a complete answer to the rationality problem (up to stable $k$-equivalence) for norm one tori $T=R_{K\mathrm{/}k}^{(1)}({\mathbb{G}}_m)$ of $K\mathrm{/}k$ whose Galois closures $L\mathrm{/}k$ are $A_5\simeq {\mathrm{P}\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_4)$ and ${\mathrm{P}\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_8)$ extensions. In particular, we prove that $T$ is stably $k$-rational for $G=\mathrm{G}\mathrm{a}\mathrm{l}(L\mathrm{/}k)\simeq {\mathrm{P}\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_8)$, $H=\mathrm{G}\mathrm{a}\mathrm{l}(L\mathrm{/}K)\simeq (C_2{)}^3$ and $H\simeq (C_2{)}^3⋊C_7$ where $C_n$ is the cyclic group of order $n$ by using GAP computations with the aid of PARI/GP. Based on the result, we conjecture that $T$ is stably $k$-rational for $G\simeq {\mathrm{P}\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_{2^d})$, $(C_2{)}^d\le H\le (C_2{)}^d⋊C_{2^d-1}$. Some other cases $G\simeq A_n$, $S_n$, ${\mathrm{G}\mathrm{L}}_n({\mathbb{F}}_{p^d})$, ${\mathrm{S}\mathrm{L}}_n({\mathbb{F}}_{p^d})$, ${\mathrm{P}\mathrm{G}\mathrm{L}}_n({\mathbb{F}}_{p^d})$, ${\mathrm{P}\mathrm{S}\mathrm{L}}_n({\mathbb{F}}_{p^d})$ and $H⪇G$ are also investigated for small $n$ and $p^d$.