Let $E$ be a known, "secure" curve, defined over a field $\mathbb{F}_q$ where $q$ is either a prime $\geq 5$ or a power of $2$. Denote by $n$ the amount of rational points of $E$.
Consider $E/\mathbb{F}_{q^2}$, the same curve but defined over the 2-degree extension field. It is clear that any $E(\mathbb{F}_q)$ is a subgroup of $E(\mathbb{F}_{q^2})$, so by Lagrange, $m := |E(\mathbb{F}_{q^2})| = nl$. Actually, with Weil's conjectures, one has $m = n (2q + 2 - n)$.
With this we see that the discrete logarithm in the extended curve is controlled by the largest prime factor of $n$ or $2q + 2 - n$, so not much bits of security are gained by considering this curve against the known attacks on the discrete logarithm (for instance, if $n$ is the largest prime factor of $m$, literally no security is gained). But that's fine for my purposes.
My question is; is the extended structure useful to the attacker, e.g., is it possible for the curve $E(\mathbb{F}_{q^2})$ to be less secure than $E(\mathbb{F}_q$)? My intuition says no, because it that was the case, then one embeds any DLOG instance on the extended curve, and solve that. But there is security degradation when higher-degree extensions are used, by means of discrete log transfers! (e.g. see 1 and 2)
