 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 $R_{g}f\in I$, and evaluation at the identity gives $f(g)=0$. Thus $g\in H$, and ${\mathrm{Stab}}_G(W)=H$.

Put $d=\dim W$ and take $V={\bigwedge }^{d}U$. The linedetermines $W$ uniquely, so its stabilizer is the stabilizer of $W$, namely $H$.

Solved by gpt-5.6-sol high.