Letbe an algebra that is graded by a groupG. Then the Hochschild and cyclic homologies ofShave canonical decompositions with components labeled by the setT(G) of conjugacy classes ofG:for Hochschild homology and similarly for cyclic homology. In this article, we describe the componentsof Hochschild homology in the case whereSis stronglyGgraded.The description is given in terms of a spectral sequencewhereHq(R,Sg) is the Hochschild homology of the identity componentR=SeofSwith coefficients in the bimoduleSgandHp(CG(g),.) is the group homology of the centralizerCG(g) ofginG. IfRis a separable algebra then the spectral sequence degenerates and yields an isomorphism