Let $N,G,G^\prime$ be three affine algebraic group varieties (i.e geometrically reduced in the sense of J. Milnes) defined over a separably closed field $K$.
Suppose that we have the following exact sequence:
$$e\rightarrow N\rightarrow G\rightarrow G^\prime\rightarrow e$$


which means $N$ is a subgroup variety of $G$, and the $G\rightarrow G^\prime$ is faithfully flat. 
My question is considering the $K$-points, do we still have the following exact sequence:
$$e\rightarrow N(K)\rightarrow G(K)\rightarrow G^\prime(K)\rightarrow e?$$



k-points of an exact sequence of algebraic varieties
Yes, this is true. A group scheme over a field is smooth if and only if it is geometrically reduced, so the hypotheses ensure that $N$ is smooth. You can even allow $G$ and $G'$ to be arbitrary group schemes. 


The map $G \rightarrow G'$ is an $N$-torsor, so it is a smooth morphism. This, for any $K$-point $g' \in G'(K)$, the fiber over $g'$ is a smooth $K$-scheme. Since $K$ is separably closed, every smooth $K$-scheme has a $K$-point. (This is an instance of the general fact that smooth morphisms of schemes acquire sections after passing to an étale cover of the base). 


