Let $\mathbb{G}$ be a connected reductive group over $\mathbb{F}_q$, and let $G$ be the base change to an algebraic closure of the base field. Denote by $F$ the associated geometric Frobenius.

Let $T$ be an $F$-rational (i.e. $FT=T$) maximal torus of $G$. As $T$ is $F$-rational, the Frobenius $F$ acts on the finite Weyl group $W(T)$. In general, the subgroup $W(T)^F$ consisting of $F$-fixed points is a proper subgroup, and depends on the choice of $T$.

How to determine when the longest element of $W(T)$ appears in $W(T)^F$? Is there a (maybe case by case) list available?

Let $\mathbb{G}$ be a connected reductive group over $\mathbb{F}_q$, and let $G$ be the base change to an algebraic closure of the base field. Denote by $F$ the associated geometric Frobenius.

Let $T$ be an $F$-rational (i.e. $FT=T$) maximal torus of $G$. As $T$ is $F$-rational, the Frobenius $F$ acts on the finite Weyl group $W(T)$. In general, the subgroup $W(T)^F$ consisting of $F$-fixed points is a proper subgroup, and depends on the choice of $T$.

How to determine when the longest element of $W(T)$ appears in $W(T)^F$? Is there a (maybe case by case) list available?

As commented below, the longest element is with respect to a Borel subgroup $B$ containing $T$: Then the longest element is the element corresponding to the open piece in $G=\cup_{w\in W(T)}BwB$.