Can we define isogeny map between ordinary elliptic curve over field?

Question

Let $k$ be a field, and $E_1, E_2$ be ordinary elliptic curves over $k$.

Given isogeny map $\phi:E_1\times_k \operatorname{Spec}\overline{k}\to E_2\times_k \operatorname{Spec}\overline{k}$ over $\overline{k}$, can we define isogeny map over $k$?

In paticular, $\operatorname{End}_{k}(E)=\operatorname{End}_{\overline{k}}(E)$?

Especially I want to know the case $k$ is finite filed.