Assume $G$ is the Klein four group $G=\{1,\sigma_1,\sigma_2,\sigma_3\}$.

Let $G$ act on $X=\mathbb{A}^2\times\mathbb{P}^1$ via:

$\sigma_1.(x,y,[\lambda:\mu])=(-x,y,[\lambda:-\mu])$ and $\sigma_2.((x,y,[\lambda:\mu])=(x,-y,[\mu:\lambda])$.

Is the quotient $X/G$ some well known variety? What are its singularities? Can one understand the fiber over $(0,0)$ of the map $X/G \rightarrow \mathbb{A}^2/G$ induced from the projection on the first factor?

I tried to do some computations: remove $0,\infty \in \mathbb{P}^1$, let $z$ be the coordinate on the rest. Then I tried to find $\mathbb{C}[x,y,z,\frac{1}{z}]^G$ and found a lot of invariants with a lot of relations, for example: $x^2$, $y^2$, $(z+\frac{1}{z})x$, $(z-\frac{1}{z})xy$, $z^2+\frac{1}{z^2}$ etc. But I don't see what variety this ring of invariants describes. Maybe it is well known?

Assume $G$ is the Klein four group $G=\{1,\sigma_1,\sigma_2,\sigma_3\}$.

Let $G$ act on $X=\mathbb{A}^2\times\mathbb{P}^1$ via:

$$\sigma_1\cdot(x,y,[\lambda:\mu])=(-x,y,[\lambda:-\mu]) \text{ and } \sigma_2 \cdot ((x,y,[\lambda:\mu])=(x,-y,[\mu:\lambda]).$$

Is the quotient $X/G$ some well known variety? What are its singularities? Can one understand the fiber over $(0,0)$ of the map $X/G \rightarrow \mathbb{A}^2/G$ induced from the projection on the first factor?

I tried to do some computations: remove $0,\infty \in \mathbb{P}^1$, let $z$ be the coordinate on the rest. Then I tried to find $\mathbb{C}[x,y,z,\frac{1}{z}]^G$ and found a lot of invariants with a lot of relations, for example: $x^2$, $y^2$, $(z+\frac{1}{z})x$, $(z-\frac{1}{z})xy$, $z^2+\frac{1}{z^2}$ etc. But I don't see what variety this ring of invariants describes. Maybe it is well known?