Let U be a smooth connected unipotent algebraic group over an algebraic closure $k$ of $\mathbf{F}_p$. Let $U' = [U,U]$ be the derived (= commutator) subgroup, and assume it is central in $U$. 
Let $V$ be a subgroup of $U(k)/U'(k)$.

Is there a subgroup $H$ of $U(k)$ such that

(a) $\pi(H) = V$, 

(b) $H\cap U' =[H,H]$?         

(Note that $H\cap U'$ must contain $[H,H]$, which depends only on $V$.) The case of interest is $U = \mathcal{U}/\mathcal{D}^2(\mathcal{U})$, where $\mathcal{U}$ is the unipotent radical of a Borel subgroup of a connected semisimple $k$-group and {$\mathcal{D}^i(\mathcal{U})$} denotes its derived series.