Do the absolute roots restricting to a given root form a Galois orbit?

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?

To answer your question, it's enough to point out that every root (in a reduced root system such as $\Phi$) plays the role of simple root in some basis $\Delta$. Note however that some absolute roots may be trivial (i.e., "zero") on restriction to $S$ and thus not ropts in the (possibly non-reduced) root system $_k\Phi$.
On the other hand, limiting the discussion of examples to the quasisplit case is artificial though convenient for exposition as in Conrad's notes. It's most helpful to go back to the more detailed treatment in the 1965 paper by Borel and Tits here: see especially $\S3-\S5$, followed by discussion in $\S6$ of the action by the Galois group $\Gamma$ of the separable closure over $k$ in an algebraic closure of an arbitrary field $k$. (The extreme case of an anisotropic $G$ over $k$ is left aside, since the classification of possibilities depends very much on $k$. The Borel-Tits structure theory for a connected reductive $G$ over a field is only uniform in the $k$-isotropic situation, i.e., when $S$ is nontrivial. And of course the case when $G$ is nontrivial and semisimple is the essential one.)