Action of $N(H)/H$ on the colors of a spherical homogeneous space $G/H$ Let $G$ be a semisimple group over $\mathbb C$, and $X=G/H$ be a spherical homogeneous space of $G$.
Let $T\subset B\subset G$ be a maximal torus and a Borel subgroup.
Let $S=S(G,T,B)$ denote the corresponding set of simple roots.
Let ${\mathcal{P}}(S)$ denote the set of subsets of  $S$.
Let  $M$ denote the weight lattice of $X$, and set  $N:={\rm Hom}(M,\mathbb Z)$. Let $\mathcal D$ denote the set of colors of $X$.
We have maps $\rho\colon \mathcal D\to N$ and ${\varsigma}\colon\mathcal D\to{\mathcal{P}}(S)$.
Here ${\varsigma}(D)$ for $D\in\mathcal D$ is the set of simple roots $\alpha\in S$ such that the corresponding minimal parabolic subgroup $P_\alpha\supset B$ moves the color  $D$.
Thus we obtain a map
$$ {\varsigma}\times\rho\colon\ \mathcal D\ \longrightarrow\  {\mathcal{P}}(S)\times N.$$
This map need not be injective, but by Proposition 3.2.3 of
Losev's paper "Uniqueness property for spherical homogeneous spaces" each of its fibers has $\le 2$ elements.
Now consider the group ${{\rm Aut}}_G(X)=\mathcal N_G(H)/H$, this group acts on $\mathcal D$.
One can easily see that ${{\rm Aut}}_G(X)$ acts on the fibers of ${\varsigma}\times\rho$.

Question 1. Is it true that ${{\rm Aut}}_G(X)$ acts transitively on each fiber of  ${\varsigma}\times\rho$ ?
Question 2. In particular, is it true that if $\mathcal N_G(H)=H$, then then the map  ${\varsigma}\times\rho$ is injective?

 A: The answer to both questions is "yes" as was communicated to me by Losev.
The main reason is that already $\varsigma$ is almost injective. More precisely, the sets $\varsigma(D)$ are always non-empty and they are pairwise disjoint except for a very well controlled situation.
Assume $I:=\varsigma(D_1)\cap\varsigma(D_2)\ne\emptyset$. Then $D_1$ and $D_2$ are moved by the same simple root $\alpha$. Luna has shown that this can only happen if $\alpha$ is also a spherical root. He showed further that then all elements of $\rho(D_i)$ are simple roots which are spherical. The Luna axioms easily imply $I=\{\alpha\}$. 
So the only way for $\varsigma(D_1)=\varsigma(D_2)$ is when $\varsigma(D_1)=\varsigma(D_2)=\{\alpha\}$. At this point one has to look at $\rho$. If $\rho(D_1)=\rho(D_2)$ then both are equal to $\frac12\alpha^\vee$ (since their sum is $\alpha^\vee$). This implies that it is possible to replace $\alpha$ by $2\alpha$ in the homogeneous spherical datum for $G/H$. This modified datum corresponds to a subgroup $\tilde H$ in which $H$ has index two. So $G/H\to G/\tilde H$ is obtained an automorphism $\phi$ of order $2$. Since $D_1$ and $D_2$ are mapped to the same color in $G/\tilde H$ they are swapped by $\phi$.
As an extra bonus one gets that $\phi$ interchanges only $D_1$ and $D_2$ and leaves all other colors fixed.
