  In a recent paper the authors Beliakova, Blanchet and Gainutdinov have shown
that the modified trace on the category $H$-pmod of the projective modules
corresponds to the symmetrised integral on the finite dimensional pivotal Hopf
algebra $H$. We generalize this fact to the context of $G$-graded categories
and Hopf $G$-coalgebra studied by Turaev-Virelizier. We show that the
symmetrised $G$-integral on a finite type pivotal Hopf $G$-coalgebra induces a
modified trace in the associated $G$-graded category.
