$\newcommand{\sss}{{\rm ss}} \newcommand{\ssc}{{\rm sc}} \newcommand{\tor}{{\rm tor}} \newcommand{\X}{{\sf X}} \newcommand{\Q}{{\mathbb Q}} \newcommand{\qed}{{$\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 $$\varphi_{12}\colon \pi_1(G_1,T_1)\overset\sim\longrightarrow \pi_1(G_2,T_2).$$ 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.