$\newcommand{\sss}{{\rm ss}}
\newcommand{\ssc}{{\rm sc}}
\newcommand{\tor}{{\rm tor}}
\newcommand{\X}{{\sf X}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\qed}{{$\blacksquare$}}
$Let $G$ be a (connected) reductive group over a field $K$.
Let $\Gamma={\rm Gal}(K^s/K)$ denote the absolute Galois group of $K$.
Notation. We write $G^\sss=[G,G]$ for the derived group of $G$ (it is semisimple),
and $G^\tor=G/G^\sss$ (it is a $K$-torus).
We denote by $G^\ssc$ the universal cover of $G^\sss$ (it is simply connected)
and consider the composite homomorphism
$$ \rho\colon\, G^\ssc\twoheadrightarrow G^\sss\hookrightarrow G.$$
Let $T\subseteq G$ be a maximal torus (defined over $K$).
Set $T^\ssc=\rho^{-1}(T)\subseteq G^\ssc$.
Let $\X_*(T)$ denote the cocharacter group of $T$ (over $K^s$).
We set
$$\pi_1(G,T)=\X_*(T)/\rho_*\X_*(T^\ssc).$$
The Galois group $\Gamma$ naturally acts on $\pi_1(G,T)$.
Proposition 1. For any two maximal tori $T_1,T_2\subseteq G$,
there is a canonical isomorphism of $\Gamma$-modules
$$\varphi_{12}\colon\, \pi_1(G_1,T_1)\overset\sim\longrightarrow \pi_1(G_2,T_2).$$
Moreover, for any third maximal torus $T_3\subseteq G$, we have
$$\varphi_{13}=\varphi_{23}\circ\varphi_{12}$$
with the obvious notations.
We need a lemma.
Lemma 2. Let
$$ 1\to G'\overset i \longrightarrow G\overset j \longrightarrow G''\to 1$$
be a short exact sequence of reductive $K$-group.
Let $T\subseteq G$ be a maximal torus.
Set
$$T'=i^{-1}(T)\subseteq G',\quad\ T''=j(T)\subseteq G''.$$
Then the natural sequence
\begin{equation}\label{e:*}
0\to \pi_1(G',T')\to \pi_1(G,T)\to \pi_1(G'',T'')\to 0\tag{1}
\end{equation}
is exact.
Idea of proof. From the commutative diagram with exact rows
$$\require{AMScd}
\begin{CD}
1 @>>> T^{\prime\,\ssc} @>>> T^\ssc @>>> T^{\prime\prime\,\ssc} @>>> 1\\
@. @VVV @VVV @VVV \\
1 @>>> T^{\prime} @>>> T @>>> T^{\prime\prime} @>>> 1
\end{CD}
$$
we obtain a commutative diagram with exact rows and injective vertical arrows
$$\require{AMScd}
\begin{CD}
0 @>>>\X_*( T^{\prime\,\ssc}) @>>> \X_*(T^\ssc) @>>> \X_*(T^{\prime\prime\,\ssc}) @>>> 0\\
@. @VVV @VVV @VVV \\
0 @>>> \X_*(T^{\prime}) @>>> \X_*( T) @>>> \X_*(T^{\prime\prime}) @>>> 0
\end{CD}
$$
Now the exactness of \eqref{e:*} follows from the snake lemma.
Now let $T_1,T_2\subseteq G$ be two maximal tori (defined over $K$).
Then there exists an element $g\in G(K^s)$ such that
\begin{equation}\label{e:2}
T_2=g \cdot T_1\cdot g^{-1}.\tag{2}
\end{equation}
We obtain an isomorphism
$$g_*\colon \pi_1(G,T_1)\overset\sim\longrightarrow \pi_1(G,T_2).$$
Lemma 3.
The isomorphism $g_*$ above does not depend on the choice of $g$ satisfying \eqref{e:2}.
Proof. Let $g'\in G(K^s)$ be another element satisfying \eqref{e:2}.
Then
$$ g^{-1}g'\cdot T_1 \cdot ( g^{-1}g')^{-1} = T_1.$$
Let $N_1$ denote the normalizer of $T_1$ in $G$.
Set $n=g^{-1} g'$.
Then $n\in N_1(K^s)$ and $g'=gn$,
whence
$$ g'_*=g_*\circ n_*.$$
By Lemma 6 below, the group $N_1(K^s)$, when acting on $\X_*(T_1)$ and $\X_*(T_1^\ssc)$ by conjugation,
acts trivially on $\pi_1(G,T_1)$, which completes the proof of Lemma 3.
Corollary 4.
The isomorphism $g_*$ above preserves the action of the Galois group $\Gamma={\rm Gal}(K^s/K)$.
Proof.
Let
$$x_1\in\pi_1(G,T_1),\quad x_2= g_*(x_1)\in\pi_1(G,T_2),\quad \gamma\in\Gamma.$$
Then
$$^\gamma\!x_2=(\,^\gamma\! g)_*(\,^\gamma\! x_1).$$
We obtain from \eqref{e:2} that
$$^\gamma T_2={}^\gamma\!g \cdot {}^\gamma T_1\cdot {}^\gamma\!g^{-1}. $$
Since $T_1$ and $T_2$ are defined over $K$, we have $^\gamma T_1=T_1$ and $^\gamma T_2=T_2$.
Therefore, we obtain that
$$ T_2={}^\gamma\!g \cdot T_1\cdot {}^\gamma\!g^{-1}. $$
Comparing with \eqref{e:2}, we see from Lemma 3 that $({}^\gamma\! g)_*=g_*$, that is,
$$^\gamma\!x_2=g_*({}^\gamma\! x_1).$$
Thus our isomorphism $g_*$ preserves the $\Gamma$-action, as desired.
Definition 5. A toric-sc resolution of a reductive $K$-group $G$ is a short exact sequence of reductive $K$-groups
\begin{equation}\label{e:***}
1\to S\to H\to G\to 1\tag{3}
\end{equation}
where $S$ is a $K$-torus and $H^\sss$ is simply connected.
The flasque resolutions of Colliot-Thélène, Résolutions flasques des groupes linéaires connexes,
J. Reine Angew. Math. 618 (2008), 77–133,
and also the $L/K$-free resolutions of
Borovoi, The defect of weak approximation for a reductive group over a global field,
are special cases of toric-sc resolutions.
Any reductive $K$-group $G$ admits a toric-sc resolution; for proofs, see Proposition-Definition 3.1 of
Colliot-Thélène's paper or Proposition 2.10 of my preprint.
Lemma 6. Let $T\subseteq G$ be a maximal torus of a reductive group over a field $K$.
Write $N$ for the normalizer of $T$ in $G$.
Then $N(K^s)$, when acting on $\X_*(T)$ and $\X_*(T^\ssc)$, acts on $\pi_1(G,T)$ trivially.
Proof. Choose a toric-sc resolution \eqref{e:***} of $G$
and consider the corresponding short exact sequence of fundamental groups
\begin{equation}\label{e:4}
0\to \X_*(S)\to \pi_1(H,T_H)\to\pi_1(G,T)\to 0\tag{4}
\end{equation}
where $T_H\subseteq H$ is the preimage of $T$ in $H$.
Since $H^\sss$ is simply connected, we have a canonical isomorphism
$$\pi_1(H,T_H)\overset\sim\longrightarrow \X_*(H^\tor).$$
Let $C_H=Z(H)^0$ denote the radical of $H$ (the identity component of the center $Z(H)$ of $H$).
Then the natural homomorphism $C_H\to H^\tor$ is an isogeny of $K$-tori, and so it induces an isomorphism
$$ \X_*(C_H)\otimes \Q\overset\sim\longrightarrow \X_*(H^\tor)\otimes\Q.$$
Let $N_H$ denote the normalizer of $T_H$ in $H$.
We consider the Weyl group $W=N_H/T_H\cong N/T$,
which naturally and compatibly acts on $\pi_1(H,T_H)$ and on $\pi_1(G,T)$
Since the torus $C_H$ is central in $H$, we see that $N_H$ acts trivially
on $C_H$ and on $ \X_*(C_H)\otimes \Q$.
Therefore, it acts trivially on $\X_*(H^\tor)\otimes \Q$ and on $ \X_*(H^\tor)$.
Thus $W$ acts trivially on $\pi_1(H,T_H)=\X_*(H^\tor)$, and we see
from \eqref{e:4} that $W$ acts trivially on $\pi_1(G,T)$.
This completes the proofs of Lemma 6, Lemma 3, Corollary 4, and Proposition 1.
