I am trying to understand the following definition from Milne's Algebraic Groups. Let $G$ be an algebraic group over $k$ and let $k^s$ be the separable closure of $k$. Let $\pi_0(G)$ be the etale $k$-scheme of connected components of $G$ and let $G\to \pi_0(G)$ be the canonical morphism of algebraic schemes. Taking the $k^s$ points of this morphism, one gets $G(k^s)\to \pi_0(G_{k^s})$. Milne then claims that because the identity component $G^\circ$ is normal in $G$, then there exists a unique group structure on $\pi_0(G_{k^s})$ such that the map $G(k^s)\to \pi_0(G_{k^s})$ is a group homomorphism respecting the Galois action, which would then make $\pi_0(G)$ an etale group scheme over $k$.
Does this mean that the map $G(k^s)\to \pi_0(G_{k^s})$ is a surjection? And the fibres of this map are the connected components of $G(k^s)$ (topology induced from $G_{k^s}$) and $G^\circ (k^s)$ is the connected component of the identity in $G(k^s)$? If so, why are these true?
I know that the map $G_{k^s}\to \pi_0(G_{k^s})$ is surjective, the fibres are the connected components of $G_{k^s}$. I am not sure of the situation when $G_{k^s}$ is replaced by $G(k^s)$. Could $G(k^s)$ be empty? For example, in the book of Qing Liu, existence of an element of $G(k^s)$ seems to require $G$ be geometrically reduced which is not generally satisfied by algebraic groups as defined by Milne.