We work with schemes over an arbitrary field $k$. Let $X$ be a scheme, and $G$ a group scheme acting on $X$. Let $Y\subseteq X$ be a locally closed subscheme. Consider the following functor $N$: for every $k$-algebra $R$, $N(R):=\{g\in G(R): gY(R)\subseteq Y(R)\}.$

Is this functor representable? I know the proof for varieties over algebraically closed fields (looking at the ideal of $Y$ gives equations for $N(k)$), but I was wondering if I could produce it as a fiber product of some sort in a more functorial way