  We consider the rigid monoidal category of character sheaves on a smooth
commutative group scheme $G$ over a finite field $k$ and expand the scope of
the function-sheaf dictionary from connected commutative algebraic groups to
this setting. We find the group of isomorphism classes of character sheaves on
$G$ and show that it is an extension of the group of characters of $G(k)$ by a
cohomology group determined by the component group scheme of $G$. We also
classify all morphisms in the category character sheaves on $G$. As an
application, we study character sheaves on Greenberg transforms of locally
finite type N\'eron models of algebraic tori over local fields. This provides a
geometrization of quasicharacters of $p$-adic tori.
