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Let $\ell$ be a prime number and let $E$ and $E'$ be $\ell$-isogenouselliptic curves defined over a finite field $k$ of characteristic $p \ne \ell$.Suppose the groups $E(k)$ and $E'(k)$ are isomorphic, but $E(K) \not \simeqE'(K)$, where $K$ is an $\ell$-power extension of $k$. In a previous work wehave shown that, under mild rationality hypotheses, the case of interest iswhen $\ell=2$ and $K$ is the unique quadratic extension of $k$. In this paperwe study the likelihood of such an occurrence by fixing a pair of 2-isogenouselliptic curves $E$, $E'$ over ${\mathbf{Q}}$ and asking for the proportion ofprimes $p$ for which $E(\mathbf{F}_p) \simeq E'(\mathbf{F}_p)$ and$E(\mathbf{F}_{p^2}) \not \simeq E'(\mathbf{F}_{p^2})$.

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