Let $ G$ be an affine algebraic group over some algebraically closed field $ K$ , and let $ H$ be a closed subgroup. Assume that $ G$ acts algebraically on an affine variety $ X$ . Assume that $ X’\subseteq X$ is a closed subvariety such that $ h(X’)=X’$ for every $ h\in H$ , and such that the following condition holds: If $ Y=G\cdot x$ is a $ G$ -orbit in $ X$ , then $ Y\cap X’$ is an $ H$ -orbit in $ X’$ ( a priori it might be a union of orbits or empty). We then have an induced algebra map $ K[X]^G\to K[X’]^H$ induced by the restriction of functions. This map is injective, due to the fact that every orbit in $ X$ intersects $ X’$ nontrivially.

Question: Is this map also surjective?

For every $ f\in K[X’]^H$ there exists a unique extension to a $ G$ -invariant function $ \tilde{f}:X\to K$ , given by $ \tilde{f}(x) = \tilde{f}(g\cdot x)$ for some $ g\in G$ such that $ g\cdot x\in X’$ (this is well defined by the conditions we have on the action). Is this map always polynomial? If so, how can one prove it? If not, what would be a counterexample? Can we prove anything more specific if we assume, for example, that $ G$ or $ H$ (or both) are reductive?