Let X be a projective variety on with a action of reductive group G. Let L be a G Linearised ample line bundle on X. Let U be a G stable open subset of X. Let $U^{ss}:=X^{ss}\cap U$. Is it true that $U^{ss}//G\subset X^{ss}//G$?

What I feel is that there is only a morphism $U^{ss}//G\rightarrow X^{ss}//G$? It may not be a subset in general.

I would appreciate an argument if i am wrong.

I would like to reformulate my question:

Let $U\hookrightarrow X$ be a G invariant open immersion. Suppose X has a Good quotient X//G. Suppose that U also has a good quotient U//G. Is it true in general that U//G is a sub-variety of X//G?