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

Let $k_0$ be a field of characteristic $0$ with algebraic closure $k$. Let $G$ be a connected reductive $k$-group, and let $Y$ be a spherical variety over $k$ (a spherical homogeneous space or a spherical embedding). Let $G_0$ be a $k_0$-model ($k_0$-form) of $G$. We give necessary and sufficient conditions for the existence of a $G_0$-equivariant $k_0$-model of $Y$.