The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, I

Let $\mathrm{\ell }$ be a prime number and let $E$ and $E^{\mathrm{\prime }}$ be $\mathrm{\ell }$-isogenous elliptic curves defined over a finite field $k$ of characteristic $p\mathrm{\ne }\mathrm{\ell }$. Suppose the groups $E(k)$ and $E^{\mathrm{\prime }}(k)$ are isomorphic, but $E(K)\simeq ̸E^{\mathrm{\prime }}(K)$, where $K$ is an $\mathrm{\ell }$-power extension of $k$. In a previous work we have shown that, under mild rationality hypotheses, the case of interest is when $\mathrm{\ell }=2$ and $K$ is the unique quadratic extension of $k$. In this paper we study the likelihood of such an occurrence by fixing a pair of 2-isogenous elliptic curves $E$, $E^{\mathrm{\prime }}$ over $\mathbf{Q}$ and asking for the proportion of primes $p$ for which $E({\mathbf{F}}_p)\simeq E^{\mathrm{\prime }}({\mathbf{F}}_p)$ and $E({\mathbf{F}}_{p^2})\simeq ̸E^{\mathrm{\prime }}({\mathbf{F}}_{p^2})$.