The problem is the following:

Let $k$ be a field and let $N,G,Q: (\operatorname{Sch}/k)_{\operatorname{fppf}} \longrightarrow \operatorname{Grps}$ be fppf-sheaves from the big fppf-site of $\operatorname{Sch}/k$ to the category of groups. Assume that we have an exact sequence of fppf-sheaves $$ e\longrightarrow N \longrightarrow G \longrightarrow Q \longrightarrow e $$ and that $N$ and $Q$ are both algebraic group schemes. Then $G$ is representable by an algebraic group scheme.

I know that after some fppf base change $Q'\longrightarrow Q$ we have an equality $$ G\times_Q Q' \cong N\times_k Q' $$ and so it becomes representable. Moreover, if $N$ is affine I can use fppf-descent obtaining the desired scheme. Nevertheless I am not sure how to deal with the general case.

This is an exercise of Milne's book on Algebraic Groups (Ex 5-10).