Let $G$ be a group scheme over a perfect field $k$. Let $G_{red} \hookrightarrow G$ be the reduced underlying scheme. It can be shown that $G_{red}$ is a closed subgroup scheme of $G$. Is there an example where $G_{red}$ is not a normal subgroup scheme of $G$?
I saw this in an exercise of a book I'm reading, and just can't think of any example.