On the Grothendieck resolution for a certain finite flat commutative group scheme of order $p^n$ over an ${\mathbb{F}}_p$-algebra

For any commutative finite flat group scheme, Grothendieck constructed an embedding into some smooth group scheme. This embedding is called the Grothendieck resolution. Let $p$ be a prime number and $n$ a positive integer. In connection with the normal basis problems in the framework of group schemes proposed by Suwa and the author, we consider the Grothendieck resolution for a certain finite flat commutative group scheme of order $p^n$ over an ${\mathbb{F}}_p$-algebra.