  For a given group $G$, we construct an invariant of flat $G$-connections on
4-manifolds from a finite type involutory quasitriangular Hopf $G$-algebra.
Hopf $G$-algebras are generalizations of Hopf algebras, equipped with gradings
by $G$. In our construction, we color the dotted components of a Kirby diagram
with elements of $G$ and employ the Hennings-type procedure. When $G$ is
finite, we also define an invariant of 4-manifolds by summing the invariants
over all flat $G$-connections.
