Suppose that $k$ is an algebraically closed field of characteristic $p > 0$ and $E/k$ is a supersingular elliptic curve equipped with a full level $N$ structure $\phi$ for some $N \ge 3$ that is prime to $p$. Let $(E^{(p^{2n})}, \phi^{(p^{2n})})$ be the $p^{2n}$-Frobenius pullback of $(E, \phi)$. Suppose that one has a $k$-isomorphism $A\colon (E, \phi) \xrightarrow{\sim} (E^{(p^{2n})}, \phi^{(p^{2n})})$. How does it follow "by Galois descent" that $(E, \phi)$ must be defined over $\mathbb{F}_{p^{2n}}$?
This is claimed on top of p. 96 of Katz-Mazur, but I cannot understand what descent theorem they are trying to apply: the extension $k/\mathbb{F}_{p^2}$ is not Galois in general, so it seems that fpqc descent should be involved, but then how does $A$ give rise to a descent datum? I guess that the only role of $\phi$ is to make some cocycle condition automatic due to rigidity of level $N$ structures.
I know that the proof I am referring to may be alternatively carried out by using $j$-invariants, but I would like to understand the reasoning intended by Katz and Mazur, since such a descent technique seems useful to know.