  We study the Tannakian properties of the category of perverse sheaves on
elliptic curves endowed with the convolution product. We establish that for
certain sheaves with unipotent local monodromy over seven points the
corresponding Tannaka group is isomorphic to $G_2$. This monodromy approach
generalizes a result of Katz on the existence of $G_2$-motives in the middle
cohomology of deformations of Beauville surfaces.
