sequence of group schemes $0\to\underline{\mathbb{Z}/p\mathbb{Z}}\to G\to\mu_p\to0$

Question

Let $p$ be a prime number and $G$ be a group scheme over a field $k$ of characteristic $0$. Assume that we have the following sequence of $0\to\underline{\mathbb{Z}/p\mathbb{Z}}\to G\to\mu_p\to0$. If we also know that $G$ has a subgroup scheme isomorphic to $\mu_p$, can we deduce that the sequence is split?

Answer 1

Let $k$ contain a $p$-th root of unity, so $\mu_p\simeq\underline{\mathbb Z/p}$. Then the following non-split sequence provides a counter-example:

$$0\to\underline{\mathbb Z/p}\to \underline{\mathbb Z/p^2}\to \underline{\mathbb Z/p}\simeq\mu_p\to0.$$