Let $G_0$ be a connected semisimple algebraic group defined over an algebraically closed field $k_0$. Let $k\supset k_0$ be a larger algebraically closed field. We write $G=G_0\times_{k_0} k$ for the base change of $G_0$ from $k_0$ to $k$.
Question. Let $H\subset G$ be a spherical subgroup of $G$ (defined over $k$). Is it true that $H$ is always conjugate to a (spherical) subgroup defined over $k_0$?
In other words, is it true that there exists an element $g\in G(k)$ and a (spherical) subgroup $H_0\subset G_0$ (defined over $k_0$) such that $$ gHg^{-1}=H_0\times_{k_0} k\ ?$$