I am studying the enlightening article "The Picard Group of $\mathcal{M}_{1, 1, S}$", written by Fulton and Olsson, but I have some problems with a proof.

**Setting**
Let $\mathcal{M}_{1, 1, k}$ denote the stack of elliptic curves over an algebraically closed field $k$ with $char(k)=3$. We have
\begin{equation*}
V = Spec(k[\lambda][1/(\lambda(\lambda - 1))]) \rightarrow \mathcal{M}_{1, 1, k} \rightarrow \mathbb{A}^{1}_{k}
\end{equation*}
where the first map (which is étale) is the Legendre family, hence induced by the ellptic curve $E_{V} \rightarrow V$
\begin{equation*}
E_{V} \colon Y^{2}Z = X(X-Z)(X - \lambda Z)
\end{equation*}
and the second map is the morphism into the coarse moduli space induced by the $j$-invariant. The composition is induced by the ring map sending $j \mapsto \frac{2^{8}(\lambda^{2}-\lambda+1)^{3}}{\lambda^{2} (1 - \lambda)^{2}}$.
Recall that in $char(k)=3$ the elliptic curves have automorphism group $\mathbb{Z}/2\mathbb{Z}$ except that for the curve with $j$-invariant equal to $0$ (which in the family corresponds to $\lambda = -1$) in which case the automorphism group is isomorphic to $G = \mathbb{Z}/4\mathbb{Z} \ltimes \mathbb{Z}/3\mathbb{Z}$.

**The problem**
I want to understand how the group $G$ acts on the localization in $\mathcal{M}_{1, 1, k}$ at the point corresponding to the curve whith automorphism group $G$. Since $V \rightarrow \mathcal{M}_{1, 1, k}$ is étale, the problem is equivalent to understanding how it acts on the localization at $\lambda = -1$ in V.

**What it is said in the article**
For comfort, we can translate and consider $\mu = \lambda + 1$ so the localization is at $\mu = 0$ and we have the morphism $Spec(k[[\mu]]) \rightarrow Spec(k[[j]])$ induced by $j \mapsto \frac{\mu^{6}}{(\mu^{4}-1)}$ (since we hare in $char(k)=3$).
Now the group $G$ acts on $k[[\mu]]$. The group sits in an exact sequence
\begin{equation*}
1 \rightarrow \{ \pm 1 \} \rightarrow G \rightarrow S_{3} \rightarrow 1
\end{equation*}
and the action on $k[[\mu]]$ factors through the action of $S_{3}$ on $k[[\mu]]$ given by the two automorphisms
\begin{equation*}
\alpha \colon \mu \mapsto -\mu
\end{equation*}
and
\begin{equation*}
\beta \colon \mu \mapsto \mu/(1-\mu) = \mu(1 + \mu + \mu^{2} + \ldots).
\end{equation*}

**My question is**: How the authors of the article determine the actions of $\alpha$ and $\beta$?

**My attempt at a solution**
I think to know how $\alpha$ acts in such a way.

First of all, if we consider a model for an elliptic curve with $j$ invariant equal to 0, like $y^{2}=x^{3} + a_{4}x + a_{6}$, with $a_{4} \ne 0$, the morphism of elliptic curves are of the form \begin{equation*} \begin{split} x &\mapsto u^{2}x' + r \\ y &\mapsto u^{3}y' \end{split} \end{equation*} and imposing the equality between the curve in $(x, y)$ and the curve in $(x', y')$, i.e. imposing the morphism being an automorphism, we obtain $u^{4}=1$ and $r^{3} + a_{4}r + (1-u^{2})a_{6}=0$. The $12$ elements $(u, r)$ which satisfy the equations are the group elements of $G$. In particular $\alpha = (i, 0)$.

To see how $\alpha$ acts: let $Spec(k[x, y])/(y^{2} = x^{3} + a_{2}x^{2} + a_{4}x + a_{6})) \rightarrow Spec(k)$ be a generic elliptic curve. On the coefficients $(a_{2}, a_{4}, a_{6})$, the transformation $(u, r)=(i, 0)$ acts via $(i, 0) \star (a_{2}, a_{4}, a_{6}) = (i^{2}a_{2}, i^{4}a_{4}, i^{6}a_{6})$. Since we are interested at the point $(0, 1, 1)$ we consider the curve $y^{2} = x^{3} + \mu x^{2} + x + 1$ on which the action is $(i, 0) \star (\mu) = i^{2} \mu = - \mu$ and taking the localization at $\mu=0$ is exactly what we want to see the action of $\alpha= (i, 0)$ at the point $\lambda = -1$ (i.e. $\mu=0$).

My problem is that the previous argument cannot be applied to $\beta$.

Can somone help me? (maybe also with a simpler way to compute the action of $\alpha$)

Thanks