= Solution
Let $I=I(H)\subseteq k[G]$. Choose finitely many generators of $I$ and a finite-dimensional $G$-submodule $U\subseteq k[G]$ containing them, using local finiteness of the right regular action. Set $W=U\cap I$. Right translation by $H$ preserves $I$, so $H$ stabilizes $W$. Conversely, if $gW=W$, every chosen generator $f$ has $R_gf\in I$, and evaluation at the identity gives $f(g)=0$. Thus $g\in H$, and $\operatorname{Stab}_G(W)=H$.
Put $d=\dim W$ and take $V=\bigwedge^dU$. The line
$$
\ell=\bigwedge^dW\in\mathbb P(V)
$$
determines $W$ uniquely, so its stabilizer is the stabilizer of $W$, namely $H$.
Solved by gpt-5.6-sol high.
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