Let $R$ be a discrete valuation ring, $K$ its field of fractions, and $$ 0 \rightarrow G_K' \rightarrow G_K \rightarrow G_K'' \rightarrow 0$$ a short exact sequence of smooth $K$-group schemes of finite type. If $G_K'$ and $G_K''$ admit Neron models over $R$, does $G_K$ admit one, too?
A positive answer is claimed in 7.5/1 (b) of "Neron models" by Bosch, Lutkebohmert, and Raynaud, but the proof given there only works in the commutative case, since 10.2/1, on which the proof crucially relies, only concerns commutative groups. I therefore wonder if the statement is nevertheless also true for noncommutative groups?
I recall that a smooth $K$-group scheme $H_K$ of finite type admits a Neron model if $H_K$ extends to a smooth and separated $R$-group scheme $H$ of finite type for which the pullback map $H(S) \rightarrow H_K(S_K)$ is bijective for every smooth $R$-scheme $S$.