Consider two copies $E_1$, $E_2$ of the supersingular elliptic curve
$$
y^2 = x^3 - 1\qquad (y^2 = x^4 - 1)
$$
over a finite field $\mathbb{F}_{p^2}$ of odd characteristics $p$ such that
$$p
\equiv 2\ (\mathrm{mod}\ 3)\qquad (p \equiv 3\ (\mathrm{mod}\ 4)).
$$
In Katsura's article "*Generalized Kummer surfaces and their unirationality in characteristic p*" on 30 page it is stated that over the algebraic closure $\overline{\mathbb{F}_{p^2}}$ the Kummer surface $K = \mathrm{Kum}(E_1\times E_2)$ is birationally isomorphic to the elliptic surface
$$
y^2 = x^3 - t^4(t-1)^4\qquad (y^2 = x^3 - t^3(t-1)^3x).
$$
Is this true over $\mathbb{F}_{p^2}$? How can I see this elliptic fibration on $K$? These particular questions are very important to answer the global one Is there a purely inseparable covering $\mathbb{A}^2 \to K$ of a Kummer surface $K$ over $\mathbb{F}_{p^2}$?

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