In general one has $T^W = Z(G) \times (\mathbb{Z}/2)^r$ where $r$ is the number of direct factors in $G$ isomorphic to $\text{SO}_{2n+1}$ ($n\geq 1$) except for the case $\text{char}(k)=2$ where $r=0$. The case $\text{PGL}_2$ considered above is isomorphic to $\text{SO}_{3}$ (note $A_1=B_1$) so $r=1$ (unless $\text{char}(k)=2$ where $r=0$). Note also that $r=0$ if the derived group of $G$ is simply connected. However the converse doesnt hold, ie consider $G=PSL_3$ ($\text{char}(k)\neq 2$). References (for the compact Lie group case) are Proposition 3.2(iv) in Jackowski, McClure and Oliver - Self-homotopy equivalences of classifying spaces of compact, connected Lie groups (MSN), Section 4 in Osse - $\lambda$-structures and representation rings of compact, connected Lie groups (MSN) or Theorem 1.6 in the paper "Normalizers of maximal tori and cohomology of Weyl groups" by M. Matthey.

In general one has $T^W = Z(G) \times (\mathbb{Z}/2)^r$ where $r$ is the number of direct factors in $G$ isomorphic to $\text{SO}_{2n+1}$ ($n\geq 1$) except for the case $\text{char}(k)=2$ where $r=0$. The case $\text{PGL}_2$ considered above is isomorphic to $\text{SO}_{3}$ (note $A_1=B_1$) so $r=1$ (unless $\text{char}(k)=2$ where $r=0$). Note also that $r=0$ if the derived group of $G$ is simply connected. However the converse doesnt hold, ie consider $G=PSL_3$ ($\text{char}(k)\neq 2$). References (for the compact Lie group case) are Proposition 3.2(iv) (Update: this should be 3.2(vi)) in Jackowski, McClure and Oliver - Self-homotopy equivalences of classifying spaces of compact, connected Lie groups (MSN), Section 4 in Osse - $\lambda$-structures and representation rings of compact, connected Lie groups (MSN) or Theorem 1.6 in the paper "Normalizers of maximal tori and cohomology of Weyl groups" by M. Matthey.

In general one has $T^W = Z(G) \times (\mathbb{Z}/2)^r$ where $r$ is the number of direct factors in $G$ isomorphic to $\text{SO}_{2n+1}$ ($n\geq 1$) except for the case $\text{char}(k)=2$ where $r=0$. The case $\text{PGL}_2$ considered above is isomorphic to $\text{SO}_{3}$ (note $A_1=B_1$) so $r=1$ (unless $\text{char}(k)=2$ where $r=0$). Note also that $r=0$ if the derived group of $G$ is simply connected. However the converse doesnt hold, ie consider $G=PSL_3$ ($\text{char}(k)\neq 2$). References (for the compact Lie group case) are Proposition 3.2(iv) in Jackowski, McClure and Oliver https://mathscinet.ams.org/mathscinet-getitem?mr=1341725, Section 4 in Osse https://mathscinet.ams.org/mathscinet-getitem?mr=1471125 or Theorem 1.6 in the paper "Normalizers of maximal tori and cohomology of Weyl groups" by M. Matthey.

In general one has $T^W = Z(G) \times (\mathbb{Z}/2)^r$ where $r$ is the number of direct factors in $G$ isomorphic to $\text{SO}_{2n+1}$ ($n\geq 1$) except for the case $\text{char}(k)=2$ where $r=0$. The case $\text{PGL}_2$ considered above is isomorphic to $\text{SO}_{3}$ (note $A_1=B_1$) so $r=1$ (unless $\text{char}(k)=2$ where $r=0$). Note also that $r=0$ if the derived group of $G$ is simply connected. However the converse doesnt hold, ie consider $G=PSL_3$ ($\text{char}(k)\neq 2$). References (for the compact Lie group case) are Proposition 3.2(iv) in Jackowski, McClure and Oliver - Self-homotopy equivalences of classifying spaces of compact, connected Lie groups (MSN), Section 4 in Osse - $\lambda$-structures and representation rings of compact, connected Lie groups (MSN) or Theorem 1.6 in the paper "Normalizers of maximal tori and cohomology of Weyl groups" by M. Matthey.