Let $k$ be a field and $X$ a proper $k$-scheme. It is a theorem of Murre and Oort that the Picard functor is representable by a $k$-group scheme $\operatorname{Pic}_{X/k}$ which is locally of finite type. Thus we can talk about $\operatorname{Pic}_{X/k}^0$, the connected component of the origin. It is a characteristic subgroup scheme.
My question is: Is the reduced closed subscheme $(\operatorname{Pic}_{X/k}^0)_{\operatorname{red}}$ a subgroup scheme of $\operatorname{Pic}^0_{X/k}$? What if $X$ is normal, or even smooth?
It is a general fact about group schemes $G$ which are locally of finite type over a perfect field, that the reduced closed subscheme $G_{\operatorname{red}}$ is in fact a subgroup scheme of $G$. This can fail if the base field is imperfect. So my question is really a question about imperfect base fields. I was looking through the 'standard' references, but couldn't find a discussion of this question.
