Let $S$ be a maximal split torus of a connected, reductive group $G$. Let $P_0$ be a minimal $k$-parabolic containing $S$, $T$ a maximal torus of $P_0$ which is defined over $k$ and contains $S$, and $B$ a Borel subgroup contained in $P_0$ and containing $T$.

The choice of $P_0$ and $B$ determine simple roots $_k\Delta$ and $\Delta$ for $_k\Phi = \Phi(G,S)$ and $\Phi = \Phi(G,T)$.

For each $a \in \space _k\Delta$, the set of $\alpha \in \Delta$ which restrict to $a$ form an orbit under the $\ast$-action of $\operatorname{Gal}(k_s/k)$. If $G$ is quasisplit, then the $\ast$-action is just the usual Galois action on characters. This is explained in section 12 of Brian Conrad's notes on reductive groups over fields.

What if we take an arbitrary $a \in \space _k\Phi$? Do the set of roots in $\Phi$ which restrict to $a$ also form a Galois orbit?