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.
I am bumping this question to the top because I now suspect I have a counterexample, namely the dihedral group of order $16$, also known as $C_2 \ltimes C_8$, where $C_n$ is cyclic of order $n$. The "obvious" lift of $C_8$ is the additive group of the $3$-truncated Witt vectors, $\mathcal{W}_3$. So the obvious thing to try is a semidirect product $\mathbb{G}_a \ltimes \mathcal{W}_3$. But $\mathbb{G}_a$ has exponent $2$, so an action of $\mathbb{G}_a$ on $\mathcal{W}_3$ must live in the $2$-torsion elements of $\mathrm{Aut}(\mathcal{W}_3)$. I compute that the space of $2$-torsion elements in $\mathrm{Aut}(\mathcal{W}_3)$ has two connnected components, one of which maps $1$ to things that are $1 \bmod 4$ and one of which maps $1$ to things which are $-1 \bmod 4$. Since $\mathbb{G}_a$ is connected, it must land in one component, but the two elements of $C_2$ live in different components.
I've made some partial progress thinking about the structure of connected unipotent groups whose $\mathbb{F}_2$ points are $C_2 \ltimes C_8$, but I've gotten stuck, so I am putting up a bounty for progress either on this case or the questions as a whole.
