For simplicity, work over an algebraically closed field of characteristic $0$. Let $$\begin{aligned} X &= \text{a smooth projective variety,} \\ G &= \text{a reductive group acting linearly on $X$,} \\ H &= \text{a finite subgroup of $G$,}\\ N(H) &= \text{the normalizer of $H$ in $G$.}\\ \end{aligned}$$ For $x \in X$, let $$\begin{aligned} \text{Stab}(x) &= \{ g \in G : g \cdot x = x \},\\ X_H &= \{ x \in X : \text{Stab$(x)$ contains $H$ }\} \end{aligned}$$

Then $G$ doesn't necessarily act on $X_H$, but $N(H)$ does, so we can look at the $N(H)$-stable and semi-stable points of $X_H$ in the sense of GIT.

**Questions**
(1) Are the $N(H)$-semi-stable points of $X_H$ automatically $G$-semi-stable points of $X$? Ditto for stable points. (This seems unlikely, but I don't know a counterexample.)

(2) If not, is there some simple criterion for determining the points $x\in X_H$ such that $x$ is $G$-stable when viewed as a point of $X$?

If it helps, the specific situation I have is: $$\begin{aligned} X &= \mathbb P^N \\ G &= \text{PGL}_k~\text{or SL}_k \\ N(H) &= \text{group of diagonal matrices in $G$.}\\ \end{aligned}$$ In particular, $N(H)$ is a maximal torus.