On the graded center of graded categories

We study the G-centers of G-graded monoidal categories where G is an arbitrary group. We prove that for any spherical G-fusion category C over an algebraically closed field such that the dimension of the neutral component of C is non-zero, the G-center of C is a G-modular category. This generalizes a theorem of M. M\"uger corresponding to G=1. We also exhibit interesting objects of the G-center.