I am following up on a question asked here: Galois action on the character group of a torus
Premise: Let $F$ be a (nonarchimedean local) field and let $T$ be an $F$-torus in a linear algebraic group $G$ (also defined over $F$). As the answer in the linked question indicates, there is a natural action of the absolute Galois group $\Gamma = \mathrm{Gal}(F_s/F)$ on the character lattice $X = X^*(T)$ of the torus $T$. Rational conjugacy classes of maximal tori in $G$ are then parameterised by the 1-cohomology set $H^1(\Gamma, \mathrm{Norm}_G(T)(F_s))$.
In section 13.2.1 of Springer's Linear Algebraic Groups, he states that the action of $\Gamma$ on the lattice $X$ is via some quotient of finite index, i.e. there exists some normal subgroup $\Gamma'$ of finite index in $\Gamma$ such that the action of $\Gamma$ on $X$ factors through $\Gamma / \Gamma'$.
My question is: Is there any effective way to find the finite quotient by which $\Gamma$ acts? For instance, if my group is the (split) group $GL(2)$ (in which any torus splits over an extension of degree $\le 2$), what is the subgroup of $\Gamma$ which acts trivially on $X$? if not, can I place some obvious bound on the index $[\Gamma : \Gamma']$?
