Let $E$/$K$ be an elliptic curve over a finite field $K$, and let $L/K$ be a finite Galois extension.
Define a map $T$ : $E(L) \rightarrow E(K)$, $P \mapsto \sum_{\sigma \in G_{L/K}} P^\sigma$.
In fact, it is an exercise #10.22-b) in 'The arithmetic of Elliptic Curves' by J.Silverman.
How to prove the map $T$ is surjective?
Let $L/K$ be an extension of finite fields of degree $n$.
Let $F\colon E\to E$ be the Frobenius endomorphism. Then $1+\cdots+F^{n-1}\colon E\to E$ is not constant. Indeed, $1-F^n=(1-F)(1+\cdots+F^{n-1})$ is non-constant, since the kernel is $E(L)$, which is a discrete group. Since $E$ is a proper curve, any non-constant morphism $E\to E$ must be surjective.
Now, let $x\in E(K)$ be an arbitrary point. By surjectivity, there exists some $y\in E(\overline K)$ such that $x=(1+\cdots +F^{n-1})y$. But now applying $1-F$ shows $(1-F^n)y=(1-F)x=0$, i.e., $y=F^ny$. Thus, $y\in E(L)$, as desired.