This largely repeats(EDIT: I've rewritten my argument in terms of the inverse functor, i.e., base extension, since it is clearer and more natural this way.)
Much of what is below is simply a reorganization of what Robin Chapman wrote, but let me write it anyway.
There is aTheorem: For each prime $j$-invariant-preserving$p$, the base extension functor from the category of elliptic curves over $\overline{\mathbf{F}}_p$ to the category$\mathcal{C}_{-p}$ of elliptic curves over $\mathbf{F}_{p^2}$ on which the $p^2$-Frobenius endomorphism acts as $-p$. There is another functor that does to the same but for Frobenius acting ascategory of supersingular elliptic curves over $+p$. Moreover, each functor defines$\overline{\mathbf{F}}_p$ is an equivalence of categoriesequivalence of categories.
These results hold for every $p$Proof: To show that the functor is an equivalence of categories, includingit suffices to show that the functor is full, faithful, and essentially surjective. It is faithful $2$(trivially), and full $3$. A uniform proof of the existence(because homomorphisms between base extensions of theelliptic curves in $\mathbf{F}_{p^2}$-model with$\mathcal{C}_{-p}$ automatically respect the Frobenius $-p$ is given inon each side). Essential surjectivity follows from Lemma 3.2.1 ofin
though it sounds as if from Robin Chapman's comments, it may also be somewhere in Lang, Elliptic functions. The ideawhich is to constructproved by constructing a model for one curve and to getgetting models for the others via separable isogenies. Once a model $\square$
The same holds for each elliptic curve over $\overline{\mathbf{F}}_p$ is fixed, each homomorphism over $\overline{\mathbf{F}}_p$ descends to $\mathbf{F}_{p^2}$ because it respects the Frobenius acting on each side; that is why one gets a functor. In particular, if two elliptic curves overcategory $\mathbf{F}_{p^2}$$\mathcal{C}_p$ defined analogously, but with Frobenius $-p$ have the same $j$-invariant, any isomorphism between their base extensions to $\overline{\mathbf{F}}_{p^2}$ comes from anacting as $\mathbf{F}_{p^2}$-isomorphism$+p$.
Here Here are two approaches for constructing the second functor,proving essential surjectivity for Frobenius $+p$$\mathcal{C}_p$:
If $G:=\operatorname{Gal}(\overline{\mathbf{F}}_p/\mathbf{F}_{p^2})$ and $E$ is an elliptic curve over $\mathbf{F}_{p^2}$, and $\overline{E}$ is its base extension to $\overline{\mathbf{F}}_{p^2}$, then the image of the nontrivial element under $H^1(G,\{\pm 1\}) \to H^1(G,\operatorname{Aut} \overline{E})$ gives the quadratic twist of $E$ (even when $p$ is $2$ or $3$, and even when $j$ is $0$ or $1728$). Applying this to each $E$ with Frobenius $-p$ gives the corresponding elliptic curve with Frobenius $+p$.
Use Honda-Tate theory (actually, it goes back to Deuring in this case) to find one supersingular elliptic curve over $\mathbf{F}_{p^2}$ with Frobenius $+p$, and then repeat the proof of Lemma 3.2.1 to construct the models of all other supersingular elliptic curves via separable isogenies.