...under the assumption that $E_1,E_2$ are elliptic curves over $\mathbb{F_q}$ and that there is a (surjective) isogeny $\pi:E_1\rightarrow E_2$ defined over $\mathbb{F_{q^2}}$ obeying $\pi\phi_1=-\phi_2\pi$. Here $\phi_i$ are the the $q$-th power Frobenious endomorphisms on $E_i$.
I've made some progress in what I think must be the right lines. I have shown by considering the parallegram law for $\text{deg}$ that $|E_1(\mathbb{F_q})|+\text{deg}(1+\phi_1)=2(q+1)$. And so it in fact suffices to show $\text{ker}(1+\phi_1)$ bijects with $\text{ker}(1-\phi_2)$. I tried doing this via $\pi$ as seems natural but while it is well-defined, I cannot show it is bijective. I have not used the fact $\pi$ is defined over $\mathbb{F_{q^2}}$ up to this point.
Any help greatly appreciated!