1. Every etale morphism is finite over some nonempty open set. (For instance, locally somewhere write it as a standard etale morphism $(A[t]/f(t))_{g(t)}$, then consider the open set where the norm of $g$, that is, the resultant of $f$ and $g$, is nonzero.) If a group homomorphism $H \to G$ is finite over some nonempty open set, it is finite over all translates of that set, so it is finite everywhere.
2. It is isomorphic to $\pi_1(G,\overline{1})$. In general, if $X \to Y$ is finite etale and Galois, the kernel of the map $\pi_1(Y) \to Gal(X/Y)$ is $\pi_1(X)$.2.\pi_1(X)$. 1 1. Every etale morphism is finite over some nonempty open set. (For instance, locally somewhere write it as a standard etale morphism$(A[t]/f(t))_{g(t)}$, then consider the open set where the norm of$g$, that is, the resultant of$f$and$g$, is nonzero.) If a group homomorphism$H \to G$is finite over some nonempty open set, it is finite over all translates of that set, so it is finite everywhere. 2. It is isomorphic to$\pi_1(G,\overline{1})$. In general, if$X \to Y$is finite etale and Galois, the kernel of the map$\pi_1(Y) \to Gal(X/Y)$is$\pi_1(X)\$.2.