We work over a field $k$ with $\operatorname{char}(k)=11$.

In the paper [1], Lemma 3.5, it is shown that the K3 surface $X_0$ defined as the weighted projective hypersurface of degree $12$ $$X_0=V(t_0^{12}+t_1^{12}+x^3+y^2) \subset \mathbb{P}(1, \, 1, \, 4, \, 6)$$ admits a symplectic action of a finite group of order 660, namely $L_2(11) \simeq \operatorname{PSL}(2, \, \mathbb{F}_{11})$. Moreover, such an action also preserves the elliptic fibration $X_0 \to \mathbb{P}^1$ given by $[t_0:t_1:x:y] \mapsto [t_0:t_1]$.

In Remark 3.6 of the same paper it is claimed that, since $X_0$ is supersingular with Artin invariant $1$, then it must be isomorphic to the quartic Fermat hypersurface $$Y=V(x_0^2+x_1^2+x_2^2+x_3^2) \subset \mathbb{P}^3.$$

Questions.

1. Is an explicit isomorphism between $X_0$ and $Y$ known? If so, what is its expression?
2. What is an explicit symplectic action of $\operatorname{PSL}(2, \, \mathbb{F}_{11})$ on the Fermat hypersurface $Y$? For instance, how to write down explicitly a symplectic automorphism of $Y$ having order $11$?

Any answer or reference to the existing literature will be appreciated.

References

[1] Dolgachev, Igor V.; Keum, Jonghae, (K3) surfaces with a symplectic automorphism of order 11, J. Eur. Math. Soc. (JEMS) 11, No. 4, 799-818 (2009). ZBL1185.14035.