Sufficient condition for surjectivity of a morphism of group schemes Let $G$ be a group scheme over a field $F$, and let $f:G\to G$ be a homomorphism. Written in my notes, I have the following statement:

To check surjectivity (on $F$-rational points), it suffices to show that the induced morphism $G_K\to G_K$ is surjective, for some finite extension $K/F$.

I would like a reference for this fact.
Remarks


*

*As such, this statement may be (very) wrong. Among the hypotheses that I have on $G$: it is a connected reductive algebraic group. However, the above is stated in this level of generality because I want to see what breaks down if we don't assume enough. 

*Moreover, one might need further hypotheses on the morphism $f$.

 A: Just to get this off the unaswered list.
There are two ways of interpretting your notation:

*

*By the surjectivity of $f:G_K\to G_K$ you means as a map of schemes.


*By the surjectivity of $f:G_K\to G_K$ you mean as a map on $K$-points.
Case 1
If you mean 1) then $f:G\to G$ is already surjective as a map of schemes since the diagram
$$\begin{matrix}G_K & \to & G_K\\ \downarrow & & \downarrow \\ G& \to & G\end{matrix}$$
is commutative and the vertical maps are surjective. Then, one knows, as Qing Liu points out in the comments above, that one has an exact sequence
$$1\to (\ker f)(F)\to G(F)\to G(F)\to H^1_\text{fppf}(F,\ker f)\to H^1_\text{fpp}(F,G)\to H^1_\text{fppf}(F,G)$$
where by $H^1_\text{fppf}(F,H)$ we mean the fppf cohomology of the group schemes $H$ over $\text{Spec}(F)$ (e.g. see [Poo, §6.4]) which if $H$ is smooth just agrees with normal Galois cohomology (e.g. since every fppf torsor for $H$ over $F$ then has a section over $k^\mathrm{sep}$ by [Poo, Proposition 3.5.70]).
Case 2
Case $2$ surprisingly implies Case 1. in most cases. Indeed, note that if $G(K)\to G(K)$ is surjective then $f(|G_K|)$, which is a closed subscheme of $G_K$ by [Mil, Theorem 5.39], contains $G(K)$. But, let's assume that $G$ is connected and that $K$ is infinite and that either

*

*$K$ is perfect.

*$G$ is reductive.

Then, $G_K$ is unirational and thus $G(K)$ is dense in $G_K$ (e.g. see [Mil, Theorem 17.39]) and thus $f(|G_K|)=G_K$ so that $f:G_K\to G_K$ is surjective so that we're in Case 1 again.
[Mil] Milne, J.S., 2017. Algebraic groups: The theory of group schemes of finite type over a field (Vol. 170). Cambridge University Press.
[Poo] Poonen, B., 2017. Rational points on varieties (Vol. 186). American Mathematical Soc..
