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I denote by $\mathbb{A}^1_x$ the affine line given by $y=0$ in $\mathbb{A}_2$ and by $\mathbb{A}^1_y$ the line $x=0$ in $\mathbb{A}_2$.

Then, the fixed locus of $\langle \sigma_1 \rangle$ is $\mathbb{A}^1_y \times [1:0] \cup \mathbb{A}^1_y \times [0:1]$.

The fixed locus of $\langle \sigma_2 \rangle$ is $\mathbb{A}^1_x \times \left([1,1] \cup [-1:1] \right)$.

The fixed locus of $\sigma_3 = \sigma_1 \times \sigma_2$ is $(0,0) \times [1,i] \cup (0,0) \times [1,-i]$.

There are no points of $X$ which are fixed by the whole $G$.

Let us denote by $\pi$ the quotient map : $\mathbb{A}^2 \times \mathbb{P}^1 \rightarrow (\mathbb{A}^2 \times \mathbb{P}^1)/G$.

As a consequence of the above discussion on the stablizers, one deduces that $(\mathbb{A}^2 \times \mathbb{P}^1)/G$ looks like:

_$\mathbb{A}^1 \times (\mathbb{A}^2/\mathbb{Z}_2)$ locally around any point of $\pi(\mathbb{A}^1_y (\times [1:0] \cup [0:1]) \cup \mathbb{A}^1_x \times ([1:1] \cup [1,-1])$. Put simply, the quotient variety has surface ordinary double point singularities along: $\pi(\mathbb{A}^1_y (\times [1:0] \cup [0:1]) \cup \mathbb{A}^1_x \times ([1:1] \cup [1,-1])$.

_the quotient variety has singularity of type $\mathbb{A}^3/\mathbb{Z}_2$ locally around $\pi([1,i] \cup [1,-i])$,

_the quotient variety is smooth everywhere else.

The ordinary double points can be resolved by blowing up the singular locus, that is blowing up $\pi(\mathbb{A}^1_y (\times [1:0] \cup [0:1]) \cup \mathbb{A}^1_x \times ([1:1] \cup [1,-1])$ (which is smooth).

The singular points $\pi([1:i])$ and $\pi([1:-i])$ can also be resoved by blowing them up. But note that the variety $X/G$ is not Gorenstein at the points $\pi([1:i])$ and $\pi([1,-i])$.

As far as the fiber over $(0,0)$ of the map $X/G \rightarrow \mathbb{A}^2/G$ is concerned, I think that it is the quotient of $\mathbb{P}^1$ by the involution $[\lambda, \mu] \rightarrow [\lambda, - \mu]$. This involution has two fixed points ($[0:1]$ and $[1:0]$) and the quotient is $\mathbb{P}^1$.

I denote by $\mathbb{A}^1_x$ the affine line given by $y=0$ in $\mathbb{A}_2$ and by $\mathbb{A}^1_y$ the line $x=0$ in $\mathbb{A}_2$.

Then, the fixed locus of $\langle \sigma_1 \rangle$ is $\mathbb{A}^1_y \times [1:0] \cup \mathbb{A}^1_y \times [0:1]$.

The fixed locus of $\langle \sigma_2 \rangle$ is $\mathbb{A}^1_x \times \left([1,1] \cup [-1:1] \right)$.

The fixed locus of $\sigma_3 = \sigma_1 \times \sigma_2$ is $(0,0) \times [1,i] \cup (0,0) \times [1,-i]$.

There are no points of $X$ which are fixed by the whole $G$.

Let us denote by $\pi$ the quotient map : $\mathbb{A}^2 \times \mathbb{P}^1 \rightarrow (\mathbb{A}^2 \times \mathbb{P}^1)/G$.

As a consequence of the above discussion on the stablizers, one deduces that $(\mathbb{A}^2 \times \mathbb{P}^1)/G$ looks like:

_$\mathbb{A}^1 \times (\mathbb{A}^2/\mathbb{Z}_2)$ locally around any point of $\pi(\mathbb{A}^1_y (\times [1:0] \cup [0:1]) \cup \mathbb{A}^1_x \times ([1:1] \cup [1,-1])$. Put simply, the quotient variety has surface ordinary double point singularities along: $\pi(\mathbb{A}^1_y (\times [1:0] \cup [0:1]) \cup \mathbb{A}^1_x \times ([1:1] \cup [1,-1])$.

_the quotient variety has singularity of type $\mathbb{A}^3/\mathbb{Z}_2$ locally around $\pi((0,0) \times([1,i] \cup [1,-i]))$,

_the quotient variety is smooth everywhere else.

The ordinary double points can be resolved by blowing up the singular locus, that is blowing up $\pi(\mathbb{A}^1_y (\times [1:0] \cup [0:1]) \cup \mathbb{A}^1_x \times ([1:1] \cup [1,-1])$ (which is smooth).

The singular points $\pi(50,0) \times [1:i])$ and $\pi((0,0) \times [1:-i])$ can also be resoved by blowing them up. But note that the variety $X/G$ is not Gorenstein at the points $\pi((0,0) \times [1:i])$ and $\pi((0,0) \times [1,-i])$.

As far as the fiber over $(0,0)$ of the map $X/G \rightarrow \mathbb{A}^2/G$ is concerned, I think that it is the quotient of $\mathbb{P}^1$ by $G$ which acts as its restriction on $\mathbb{P}^1$. The quotient map is a $4:1$ cover ramified in $4$ points (namely $[1:0],[0:1],[1:i],[1:-i]$). The Hurwitz formula shows that the quotient is $\mathbb{P}^1$.

I denote by $\mathbb{A}^1_x$ the affine line given by $y=0$ in $\mathbb{A}_2$ and by $\mathbb{A}^1_y$ the line $x=0$ in $\mathbb{A}_2$.

Then, the fixed locus of $\langle \sigma_1 \rangle$ is $\mathbb{A}^1_y \times [1:0] \cup \mathbb{A}^1_y \times [0:1]$.

The fixed locus of $\langle \sigma_2 \rangle$ is $\mathbb{A}^1_x \times \left([1,1] \cup [-1:1] \right)$.

The fixed locus of $\sigma_3 = \sigma_1 \times \sigma_2$ is $[1,i] \cup

The fixed locus of $G$ is $(0,0) \times [1:0] \cup (0,0) \times [0:1]$.

Let us denote by $\pi$ the quotient map : $\mathbb{A}^2 \times \mathbb{P}^1 \rightarrow (\mathbb{A}^2 \times \mathbb{P}^1)/G$.

As a consequence of the above discussion on the stablizers, one deduces that $(\mathbb{A}^2 \times \mathbb{P}^1)/G$ looks like:

_$\mathbb{A}^1 \times (\mathbb{A}^2/\mathbb{Z}_2)$ locally around any point of $\pi(\mathbb{A}^1_y \times [1:0] \cup \mathbb{A}^1_y \times [0:1] - \left((0,0) \times [0:1] \cup (0,0) \times [1:0]\right))$. Put simply, the quotient variety has surface ordinary double point singularities along: $\pi(\mathbb{A}^1_y \times [1:0] \cup \mathbb{A}^1_y \times [0:1] - \left((0,0) \times [0:1] \times (0,0) \times [1:0]\right))$.

_the quotient variety has singularity of type $\mathbb{A}^3/G$ locally around $\pi((0,0) \times [0:1] \cup (0,0) \cup [1:0])$, where $G$ acts on $\mathbb{A}^3$ by $\sigma_1.(x,y,z) = (-x,y,-z)$ and $\sigma_2.(x,y,z) = (x,-y,z)$.

_the quotient variety is smooth everywhere else.

As far as the fiber over $(0,0)$ of the map $X/G \rightarrow \mathbb{A}^2/G$ is concerned, I think that it is the quotient of $\mathbb{P}^1$ by the involution $[\lambda, \mu] \rightarrow [\lambda, - \mu]$. This involution has two fixed points ($[0:1]$ and $[1:0]$) and the quotient is $\mathbb{P}^1$.

I denote by $\mathbb{A}^1_x$ the affine line given by $y=0$ in $\mathbb{A}_2$ and by $\mathbb{A}^1_y$ the line $x=0$ in $\mathbb{A}_2$.

Then, the fixed locus of $\langle \sigma_1 \rangle$ is $\mathbb{A}^1_y \times [1:0] \cup \mathbb{A}^1_y \times [0:1]$.

The fixed locus of $\langle \sigma_2 \rangle$ is $\mathbb{A}^1_x \times \left([1,1] \cup [-1:1] \right)$.

The fixed locus of $\sigma_3 = \sigma_1 \times \sigma_2$ is $(0,0) \times [1,i] \cup (0,0) \times [1,-i]$.

There are no points of $X$ which are fixed by the whole $G$.

Let us denote by $\pi$ the quotient map : $\mathbb{A}^2 \times \mathbb{P}^1 \rightarrow (\mathbb{A}^2 \times \mathbb{P}^1)/G$.

As a consequence of the above discussion on the stablizers, one deduces that $(\mathbb{A}^2 \times \mathbb{P}^1)/G$ looks like:

_$\mathbb{A}^1 \times (\mathbb{A}^2/\mathbb{Z}_2)$ locally around any point of $\pi(\mathbb{A}^1_y (\times [1:0] \cup [0:1]) \cup \mathbb{A}^1_x \times ([1:1] \cup [1,-1])$. Put simply, the quotient variety has surface ordinary double point singularities along: $\pi(\mathbb{A}^1_y (\times [1:0] \cup [0:1]) \cup \mathbb{A}^1_x \times ([1:1] \cup [1,-1])$.

_the quotient variety has singularity of type $\mathbb{A}^3/\mathbb{Z}_2$ locally around $\pi([1,i] \cup [1,-i])$,

_the quotient variety is smooth everywhere else.

The ordinary double points can be resolved by blowing up the singular locus, that is blowing up $\pi(\mathbb{A}^1_y (\times [1:0] \cup [0:1]) \cup \mathbb{A}^1_x \times ([1:1] \cup [1,-1])$ (which is smooth).

The singular points $\pi([1:i])$ and $\pi([1:-i])$ can also be resoved by blowing them up. But note that the variety $X/G$ is not Gorenstein at the points $\pi([1:i])$ and $\pi([1,-i])$.

As far as the fiber over $(0,0)$ of the map $X/G \rightarrow \mathbb{A}^2/G$ is concerned, I think that it is the quotient of $\mathbb{P}^1$ by the involution $[\lambda, \mu] \rightarrow [\lambda, - \mu]$. This involution has two fixed points ($[0:1]$ and $[1:0]$) and the quotient is $\mathbb{P}^1$.

I denote by $\mathbb{A}^1_x$ the affine line given by $y=0$ in $\mathbb{A}_2$ and by $\mathbb{A}^1_y$ the line $x=0$ in $\mathbb{A}_2$.

Then, the fixed locus of $\langle \sigma_1 \rangle$ is $\mathbb{A}^1_y \times [1:0] \cup \mathbb{A}^1_y \times [0:1]$.

The fixed locus of $\langle \sigma_2 \rangle$ is $\mathbb{A}^1_x \times \mathbb{P}^1$.

The fixed locus of $G$ is $(0,0) \times [1:0] \cup (0,0) \times [0:1]$.

Let us denote by $\pi$ the quotient map : $\mathbb{A}^2 \times \mathbb{P}^1 \rightarrow (\mathbb{A}^2 \times \mathbb{P}^1)/G$.

As a consequence of the above discussion on the stablizers, one deduces that $(\mathbb{A}^2 \times \mathbb{P}^1)/G$ looks like:

_$\mathbb{A}^1 \times (\mathbb{A}^2/\mathbb{Z}_2)$ locally around any point of $\pi(\mathbb{A}^1_y \times [1:0] \cup \mathbb{A}^1_y \times [0:1] - \left((0,0) \times [0:1] \cup (0,0) \times [1:0]\right))$. Put simply, the quotient variety has surface ordinary double point singularities along: $\pi(\mathbb{A}^1_y \times [1:0] \cup \mathbb{A}^1_y \times [0:1] - \left((0,0) \times [0:1] \times (0,0) \times [1:0]\right))$.

_the quotient variety has singularity of type $\mathbb{A}^3/G$ locally around $\pi((0,0) \times [0:1] \cup (0,0) \cup [1:0])$, where $G$ acts on $\mathbb{A}^3$ by $\sigma_1.(x,y,z) = (-x,y,-z)$ and $\sigma_2.(x,y,z) = (x,-y,z)$.

_the quotient variety is smooth everywhere else.

As far as the fiber over $(0,0)$ of the map $X/G \rightarrow \mathbb{A}^2/G$ is concerned, I think that it is the quotient of $\mathbb{P}^1$ by the involution $[\lambda, \mu] \rightarrow [\lambda, - \mu]$. This involution has two fixed points ($[0:1]$ and $[1:0]$) and the quotient is $\mathbb{P}^1$.

I denote by $\mathbb{A}^1_x$ the affine line given by $y=0$ in $\mathbb{A}_2$ and by $\mathbb{A}^1_y$ the line $x=0$ in $\mathbb{A}_2$.

Then, the fixed locus of $\langle \sigma_1 \rangle$ is $\mathbb{A}^1_y \times [1:0] \cup \mathbb{A}^1_y \times [0:1]$.

The fixed locus of $\langle \sigma_2 \rangle$ is $\mathbb{A}^1_x \times \left([1,1] \cup [-1:1] \right)$.

The fixed locus of $\sigma_3 = \sigma_1 \times \sigma_2$ is $[1,i] \cup

The fixed locus of $G$ is $(0,0) \times [1:0] \cup (0,0) \times [0:1]$.

Let us denote by $\pi$ the quotient map : $\mathbb{A}^2 \times \mathbb{P}^1 \rightarrow (\mathbb{A}^2 \times \mathbb{P}^1)/G$.

As a consequence of the above discussion on the stablizers, one deduces that $(\mathbb{A}^2 \times \mathbb{P}^1)/G$ looks like:

_$\mathbb{A}^1 \times (\mathbb{A}^2/\mathbb{Z}_2)$ locally around any point of $\pi(\mathbb{A}^1_y \times [1:0] \cup \mathbb{A}^1_y \times [0:1] - \left((0,0) \times [0:1] \cup (0,0) \times [1:0]\right))$. Put simply, the quotient variety has surface ordinary double point singularities along: $\pi(\mathbb{A}^1_y \times [1:0] \cup \mathbb{A}^1_y \times [0:1] - \left((0,0) \times [0:1] \times (0,0) \times [1:0]\right))$.

_the quotient variety has singularity of type $\mathbb{A}^3/G$ locally around $\pi((0,0) \times [0:1] \cup (0,0) \cup [1:0])$, where $G$ acts on $\mathbb{A}^3$ by $\sigma_1.(x,y,z) = (-x,y,-z)$ and $\sigma_2.(x,y,z) = (x,-y,z)$.

_the quotient variety is smooth everywhere else.

As far as the fiber over $(0,0)$ of the map $X/G \rightarrow \mathbb{A}^2/G$ is concerned, I think that it is the quotient of $\mathbb{P}^1$ by the involution $[\lambda, \mu] \rightarrow [\lambda, - \mu]$. This involution has two fixed points ($[0:1]$ and $[1:0]$) and the quotient is $\mathbb{P}^1$.