Existence of equivariant models of spherical homogeneous spaces and other G-varieties

Let $k_0$ be a field of characteristic $0$, and fix an algebraic closure $k$ of $k_0$. Let $G$ be an algebraic $k$-group, and let $Y$ be a $G$-$k$-variety. Let $G_0$ be a $k_0$-model ($k_0$-form) of $G$. We ask whether $Y$ admits a $G_0$-equivariant $k_0$-model. If $Y$ admits a $G_\diamondsuit$-equivariant $k_0$-model for an inner form $G_\diamondsuit$ of $G_0$, then we give a Galois-cohomological criterion for the existence of a $G_0$-equivariant $k_0$-model. If $G$ is connected reductive, $G_0$ is quasi-split, and $Y$ is a spherical homogeneous space, then we show that there exists a $G_0$-equivariant $k_0$-model of $Y$ if and only if the Galois action preserves the combinatorial invariants of $Y$. Since any $k_0$-model of a connected reductive group is an inner form of a quasi-split $k_0$-model, we obtain necessary and sufficient conditions for the existence of equivariant $k_0$-models of any spherical homogeneous space.