Let $K$ be a non-trivial extension of a finite field (to fix ideas, assume $K = \mathbb{F}_{p^2}$ for a prime $p$), and $E/K$ an elliptic curve defined over $K$.
Is it possible to find a different finite field $K'$, another elliptic curve $E'/K'$ and a group isomorphism (e.g. preserving addition of points) $\phi: E/K \to E'/K'$? I am particularly interested in the case where $K'$ has a lower extension degree than $K$ (think $K' = \mathbb{F}_q$ where $q$ is prime).
My first intuition says no, because the algebraic structure of a curve changes too much when changing the finite field, but then for $K = K'$, isogenies exist, so perhaps there is some specific relation between the fields and equations that could make this work. Some kind of "cross-field" isogeny. The main application is cryptography, so I would actually be satisfied if there is such isomorphism between the largest subgroups of each curve (motivation is to implement addition in $E/\mathbb{F}_{p^2}$ with the help of addition in $E'/\mathbb{F}_p$).