Let $G$ be the base change of a connected reductive group over ${\mathbb{F}}\_q$ to $\overline{\mathbb{F}}\_q$, and let $F$ be the associated geometric Frobenius on $G$. Let $W$ be the Weyl group of a quasisplit $F$-stable maximal torus of $G$. 


By definition, $F$ acts on $W$. We know that, the $G^F$-conjugacy classes of $F$-stable maximal tori in $G$ are one to one corresponding to the $F$-conjugacy classes of $W$, and the quasisplit ones form a single class corresponding to $1\in W$. 


**Question:** Under this picture, is it true that, the $F$-stable maximal tori which are NOT contained in any $F$-stable parabolic subgroup of $G$ form a single class?


I checked standard references like Carter's book and Digne--Michel's book but couldn't find the answer; any reference containing a positive proof or counterexamples would be appreciated!



On cuspidal maximal tori of a connected reductive group
No. For example, in (split, simply connected) type $\mathrm C\_2$, there are the tori whose rational points are isomorphic to $\ker \mathrm N\_{\mathbb F\_{q^4}/\mathbb F\_{q^2}}$, and those whose rational points are isomorphic to the product of $\ker \mathrm N\_{\mathbb F\_{q^2}/\mathbb F\_q}$ with itself.


