Equivariant models of spherical varieties

Let $G$ be a connected semisimple group over an algebraically closed field $k$ of characteristic 0. Let $Y=G\mathrm{/}H$ be a spherical homogeneous space of $G$, and let $Y^{\mathrm{\prime }}$ be a spherical embedding of $Y$. Let $k_0$ be a subfield of $k$. Let $G_0$ be a $k_0$ -model ($k_0$-form) of $G$. We show that if $G_0$ is an inner form of a split group and if the subgroup $H$ of $G$ is spherically closed, then $Y$ admits a $G_0$-equivariant $k_0$-model. If we replace the assumption that $H$ is spherically closed by the stronger assumption that $H$ coincides with its normalizer in $G$, then $Y$ and $Y^{\mathrm{\prime }}$ admit compatible $G_0$-equivariant $k_0$-models, and these models are unique.