Let $E$ be an elliptic curve over an algebraically closed field $K$.
It seems that there is a bijection between the set of finite etale covers $\phi:C\to E$ with a marked point in the fiber $\phi^{-1}(O)$ and the set of finite subgroups of $E$ which goes as follows: a cover $\phi:C\to E$ with a marked point in the fiber $\phi^{-1}(O)$ may be considered as an isogeny of elliptic curves, then the kernel of the dual isogeny $\hat\phi: E\to C$ is the corresponding finite subgroup of E.
At the same time from general theory we know that such covers with marked point correspond to marked finite sets with a continuous action of $\pi^{et}_1(E,O)=\hat{\mathbb{Z}}\times\hat{\mathbb{Z}}$.
My question is: how does the first construction agree with the other one?