  We explore the notion of representation of an affine extension of an abelian
variety -- such an extension is a faithfully flat affine morphism of
$\Bbbk$-group schemes $q:G\to A$, where $A$ is an abelian variety. We
characterize the categories that arise as the category of representations of an
affine extension $q:G\to A$, generalizing the classical results of Tannaka
Duality established for affine $\Bbbk$-group schemes (that is, when
$A=\operatorname{Spec}(\Bbbk)$). We also prove the existence of a contravariant
equivalence between the category of affine extensions of a given $A$ and the
category of faithful commutative Hopf sheaves on $A$, generalizing in this
manner the well-known op-equivalence between affine group schemes and
commutative Hopf algebras. If $\mathcal H_q$ is the Hopf sheaf on $A$
associated to $q$, the category of representations of $q$ is equivalent to the
category of $\mathcal H_q$-comodules.
