This is about section 6.2 in Borel-Tits' Groupes réductifs where they define a certain $\Gamma$-action on maximal split tori, denoted as $_\Delta \gamma$, distinct from the "usual" one. (If I am not completely wrong, the new action is the one visible in Satake-Tits diagrams, i.e. the *-action in a Tits index (Tits' classification article in the Boulder proceedings), and the $[\; \cdot \; ]$-action in Satake's Classification theory of semi-simple algebraic groups).)

They say that one has the "evident relation" (equation (3))

$\gamma (_K P_\theta) = \, _K P_{\theta'}$ where $\theta' = \, _\Delta \gamma(\theta)$.

Question 1: Is this to be read "modulo conjugation"?

Namely, the $_K P_\theta$ are standard parabolic subgroups (defined in 5.12), and I do not see that the usual $\Gamma$-action stabilises these. With the notation of 5.23, I would rather have expected and be perfectly satisfied with $\gamma (_K P_\theta) \in \, _K \mathscr{P}_{\theta'}$.

Then in the end of the section, they say that whereas the $_\Delta \gamma$-action does not depend on the choice of maximal $K$-split torus $T$, one does not necessarily have that the "usual" $\Gamma$-modules $X^*(T)$ and $X^*(T')$ are isomorphic, for two maximal $K$-split tori $T$, $T'$ both stable under $\Gamma$. In a footnote they add that there is an example due to Serre where no maximal $K$-split torus is stable under $\Gamma$ at all.

Question 2: What is an example for the first assertion, and what is Serre's example?

Question 3: Do these examples disappear in certain good situations, specifically, any combination of a) $G$ semisimple b) $K|k$ Galois c) (for the first one) both $T, T'$ defined over $k$?

It might well be that some of this is trivial, in which case I apologise.

This is about section 6.2 in Borel-Tits' Groupes réductifs where they define a certain $\Gamma$-action on maximal split tori, denoted as $_\Delta \gamma$, distinct from the "usual" one. (If I am not completely wrong, the new action is the one visible in Satake-Tits diagrams, i.e. the *-action in a Tits index (Tits' classification article in the Boulder proceedings), and the $[\; \cdot \; ]$-action in Satake's Classification theory of semi-simple algebraic groups).)

They say that one has the "evident relation" (equation (3))

$\gamma (_K P_\theta) = \, _K P_{\theta'}$ where $\theta' = \, _\Delta \gamma(\theta)$.

Question: How to see this?

Namely, the $_K P_\theta$ are standard parabolic subgroups (defined in 5.12), and I do not see that the $\Gamma$-action stabilises these. With the notation of 5.23, I would rather have expected the weaker $\gamma (_K P_\theta) \in \, _K \mathscr{P}_{\theta'}$, i.e. $\gamma (_K P_\theta)$ is a parabolic subgroup conjugate to $_K P_{\theta'}$.