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 $W_2$.

Addendum: An algebraic group is a vector group precisely when it is commutative, connected and killed by $p$ (again in Serre's book). Clearly $G$ is killed by $p$ if $E$ is. Conversely, if $G$ is killed by $p$ then multiplication by $p$ induces a map $G\to H$ which is constant as $G$ is connected and $H$ is finite. Thus $G$ is a vector group precisely when $E$ is. Also vector groups are determined by their dimension and $E$ and $G$ have the same dimension.

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 $W_2$.

Addendum: An algebraic group is a vector group precisely when it is commutative, connected and killed by $p$ (again in Serre's book). Clearly $G$ is killed by $p$ if $E$ is. Conversely, if $G$ is killed by $p$ then multiplication by $p$ induces a map $G\to H/pH$ which is constant as $G$ is connected and $H$ is finite. Thus $G$ is a vector group precisely when $E$ is. Also vector groups are determined by their dimension and $E$ and $G$ have the same dimension.

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 $W_2$.

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 $W_2$.

Addendum: An algebraic group is a vector group precisely when it is commutative, connected and killed by $p$ (again in Serre's book). Clearly $G$ is killed by $p$ if $E$ is. Conversely, if $G$ is killed by $p$ then multiplication by $p$ induces a map $G\to H$ which is constant as $G$ is connected and $H$ is finite. Thus $G$ is a vector group precisely when $E$ is. Also vector groups are determined by their dimension and $E$ and $G$ have the same dimension.