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In Groupes algébriques et corps de classes Serre classifies the $2$-dimensional commutative unipotent connected algebraic groups $G$ (VII:11). With the exception of the product of the additive group with itself they are all isogenous to the Witt vector group $W_2$ so that there is an exact sequence as per above with $E=W_2$. For some of them Serre notes that the isogeny can be chosen to be separable so that $H$ is a finite group (i.e., étale group scheme) yet $G$ is not necessarily isomorphic to $W_2$.