I wish to "compute" ${{\rm Aut}}(G,X)$ for a spherical homogeneous space $X$ of $G$ in terms of the spherical datum of $X$.

First let $G$ be any algebraic group over $\mathbb C$, and let $X$ be any left $G$-variety. We define $${{\rm Aut}}(G,X):=\{(\sigma,\mu)\in{{\rm Aut}}(G)\times{{\rm Aut}}(X)\ |\ \mu(g\cdot x)=\sigma(g)\cdot\mu(x)\text{ for } g\in G,\ x\in X\}.$$ We have a canonical homomorphism $$\pi\colon{{\rm Aut}}(G,X)\to {{\rm Aut}}(G),\quad (\sigma,\mu)\mapsto \sigma.$$ Write $${{\rm Aut}}_G(X):=\ker\,\pi=\{\mu\in{{\rm Aut}}(X)\ |\ \mu(g\cdot x)=g\cdot\mu(x) \text{ for all } g\in G,\ x\in X\},$$ then we have an exact sequence $$ 1\to{{\rm Aut}}_G(X)\to{{\rm Aut}}(G,X)\to {{\rm Aut}}(G).$$

Now assume that $G$ is a connected semisimple group and that $X$ is a spherical homogeneous space of $G$, $X=G/H$. Let $N(H)$ denote the normalizer of $H$ in $G$. Then ${{\rm Aut}}_G(X)=N(H)/H$ (because $X=G/H$) and $N(H)/H$ is a group of multiplicative type (because $X$ is spherical). This group $N(H)/H$ can be computed in terms of the spherical datum of $X$, see Friedrich Knop's answer.

Set ${{\rm Aut}}_X(G)={{\rm im}}\, \pi\subset {{\rm Aut}}(G)$. We compute ${{\rm Aut}}_X(G)$. Consider the homomorphism $ i\colon G\to {{\rm Aut}}(G,X)$ sending $a\in G$ to the "inner automorphism" $i_a\in {{\rm Aut}}(G,X)$, where $$ i_a(g,x)=(aga^{-1}, a\cdot x).$$ Clearly $\pi(i_a)={\rm inn}(a)\in {{\rm Aut}}(G)$, hence ${{\rm Aut}}_X(G)\supset {{\rm Inn}}(G)$. Write $${{\rm Out}}(G)={{\rm Aut}}(G)/{{\rm Inn}}(G),\quad {{\rm Out}}_X(G)={{\rm Aut}}_X(G)/{{\rm Inn}}(G).$$ Using Losev's uniqueness theorem and ideas of Akhiezer and Cupit-Foutou, one can describe the subgroup ${{\rm Out}}_X(G)\subset {{\rm Out}}(G)$ as follows: the group ${{\rm Out}}(G)$ acts on the Dynkin diagram and on the based root datum of the semisimple group $G$, in particular, on the character group ${{\mathcal{X}}}(B)$ of a Borel subgroup $B$ of $G$; then ${{\rm Out}}_X(G)$ is the subgroup of ${{\rm Out}}(G)$ consisting of the elements that preserve the spherical datum of $X$ when acting on ${{\mathcal{X}}}(B)$.

A pinning $p$ of $G$ defined a splitting $s_p\colon {{\rm Out}}(G)\to {{\rm Aut}}(G)$, and thus a splitting ${{\rm Out}}_X(G)\to {{\rm Aut}}_X(G)$. Thus we know ${{\rm Aut}}_X(G)$, it is a semidirect product ${{\rm Inn}}(G)\rtimes{{\rm Out}}_X(G)$. We know also ${{\rm Aut}}_G(X)=N(H)/H$.

I wish to understand the extension $$1\to {{\rm Aut}}_G(X)\to {{\rm Aut}}(G,X)\to {{\rm Aut}}_X(G)\to 1.$$ Using the splitting $s_p\colon {{\rm Out}}_X(G)\to {{\rm Aut}}_X(G)$, we obtain an extension $$1\to {{\rm Aut}}_G(X)\to\mathfrak{G}\to{{\rm Out}}_X(G)\to 1.$$ This last extension defines a cohomology class $$\eta_X\in H^2({{\rm Out}}_X(G), {{\rm Aut}}_G(X)).$$

Question 1.Are there examples for which this extension does not split, that is, $\eta_X\neq 0$ ?

Question 2.How can one compute $\eta_X$ in terms of the spherical datum of $X$ ?

EDIT (based on a comment of YCor). Set ${\rm Inn}(G,H)={\rm im}\,i\subset{\rm Aut}(G,X)$, which is a normal subgroup of ${\rm Aut}(G,X)$. We set ${\rm Out}(G,X)={\rm Aut}(G,X)/{\rm Inn}(G,X)$. The epimorphism $\pi\colon {\rm Aut}(G,X)\to {\rm Aut}_X(G)$ induces an epimorphism $$\bar\pi\colon {\rm Out}(G,X)\to{\rm Out}_X(G)$$ whose kernel seems to be $N(H)/(Z(G)H)$. The following question seems to be easier than Question 2.

Question 3.How can one describe the short exact sequence $$ 1\to N(H)/(Z(G)H)\to {\rm Out}(G,X)\to {\rm Out}_X(G)\to 1 $$ in terms of the spherical datum of $X$ ?