  Let $\ell$ be a prime number and let $E$ and $E'$ be $\ell$-isogenous
elliptic curves defined over a finite field $k$ of characteristic $p \ne \ell$.
Suppose the groups $E(k)$ and $E'(k)$ are isomorphic, but $E(K) \not \simeq
E'(K)$, where $K$ is an $\ell$-power extension of $k$. In a previous work we
have shown that, under mild rationality hypotheses, the case of interest is
when $\ell=2$ 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 $E$, $E'$ over ${\mathbf{Q}}$ and asking for the proportion of
primes $p$ for which $E(\mathbf{F}_p) \simeq E'(\mathbf{F}_p)$ and
$E(\mathbf{F}_{p^2}) \not \simeq E'(\mathbf{F}_{p^2})$.
