= Solution
For $G=\mathbb G_m$ and $H=\mu_2$, the quotient is
$$
\mathbb G_m\longrightarrow\mathbb G_m,
\qquad z\longmapsto z^2.
$$
It is a faithfully flat $\mu_2$-torsor, including in characteristic two where $\mu_2$ is nonreduced. It is not a Zariski torsor: a local section over any nonempty open set would put a square root of the coordinate $t$ in the function field $k(t)$, but $t$ is not a square there.
Take $V=k\oplus k$ with $z\in\mathbb G_m$ acting by $z\cdot(x,y)=(x,z^2y)$, and let $\ell=k(1,1)$. The equality $z\ell=\ell$ holds exactly when $z^2=1$, so the scheme-theoretic stabilizer is $\mu_2$.
Solved by gpt-5.6-sol high.
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