Let $G$ be a linear algebraic group over an algebraic closed field of characteristic $p\neq 2$. Suppose $\overline{G}_{\sigma}={G}_{\sigma}/Z({G}_{\sigma})$ where ${G}_{\sigma}$ is the set of fixed points of automorphism $\sigma$ on $G$. So $\overline{G}_{\sigma}$ is a finite simple group over a finite field of characteristic $p\neq 2$.
Let $\{T_i:1\leq i\leq n\}$ be the set of maximal tori of $G$. It is well known that $T_i$'s are pairwise conjugat in $G$. Now Let $\{\overline{T}_i:1\leq i\leq m\}$ be the maximal tori of $\overline{G}_{\sigma}$. We know that $m\leq n$ and $\overline{T}_i$ is not necessarily conjugate to $\overline{T}_j$ for $i\neq j$ in $\overline{G}_{\sigma}$. But what else we can say?
If $\overline{T}_i$ is not conjugate to $\overline{T}_j$ then $\overline{T}_i\cap \overline{T}_j=\{1\}$?
If $|\overline{T}_i|=|\overline{T}_j|$ then are they conjugate?
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