Let $1\to H\to E\to G\to 1$ be a short exact sequence of algebraic groups defined over an algebraically closed field $k$ of characteristic $p$. Suppose $H$ is a finite group, and $G$ and $E$ are connected. Does it follow that $G\cong E$?

[Edit: of course not in general, since, as Max points out, E=SL_n, H=Z(E) and G=PSL_n yield counterexamples]

More specifically for my purposes, if $G$ is a vector group, (i.e. $G$ is isomorphic to the direct product of $n$ copies of the additive group $G_a(k)$ of the field $k$) must $E$ be one as well? (Let $E$ also be unipotent, if it helps.)

I note that this follows when $G=G_a(k)$, the additive group of the field, since there are only two connected 1-dimensional groups and $G_m(k)$ does not surject onto $G_a(k)$.

Differentiating, one sees that $L(E)\cong L(G)$, for what it's worth.