Let $U$ be an open subset of $\mathbb P^1$ without two points (say $t=0$ and $t=\infty$) and $j: U\to \mathbb P^1$ be an open immersion. Ground field $k$ is algeraically closed. Let $G$ be the group scheme over $\mathbb P^1$ given by

$x^2 - ty^2 = 1$

I am interested in computing the stalks of the skyscraper $R^1j_*G$. It looks that the stalk at point $t=0$ must be isomorphic to the group

$\mathrm H^1_{et}(k((t)), G) \simeq \mathbb Z/2$

but cannot confirm this result.

My motivation comes from this example, but I want to apply it for the group scheme $G$.

P.S. Moreover, it looks that $j_*G \simeq G$, just by computing the stalks of the quotient for a natural map $G \to j_*G$ on $\mathbb P^1$. Cannot confirm this either.