Let $\Gamma$ be a finite group of order $p^n$. Is there necessarily a unipotent algebraic group $G$ of dimension $n$, defined over $\mathbb{F}_p$, with $\Gamma \cong G(\mathbb{F}_p)$?

I have no real motivation for this question, it just came up in conversation and no one knew the answer. There does not appear to be any sort of uniqueness to $G$; both the groups $\mathbb{Z}/p^2 \mathbb{Z}$ and $(\mathbb{Z}/p \mathbb{Z})^2$ have infinitely many lifts to unipotent groups.