Let $G$ be a linearly reductive algebraic group and $X$ be an affine $G$-variety over an algebraically closed field $\mathbb{K}$. Let $Y\subset X$ be a (closed) affine subvariety of $X$ which is also $G$-stable.

Is there any "nice" description of the invariant ring $\mathbb{K}[Y]^G$ in terms of the invariant ring $\mathbb{K}[X]^G$? Maybe something like- it's a quotient or a subring of $\mathbb{K}[X]^G$.

For my situation, $X$ is an affine space.