Full Level Structure on Some Group Schemes

We give a definition of full level structure on group schemes of the form $G\times G$, where $G$ is a finite flat commutative group scheme of rank $p$ over a $\mathbb{Z}_p$-scheme $S$ or, more generally, a truncated $p$-divisible group of height $1$. We show that there is no natural notion of full level structure over the stack of all finite flat commutative group schemes.