Clifford theory is a technique which reduces blocks of a finite group to blocks of central extensions of its quotient subgroups. The first step in this technique is reduction to the inertial subgroup. In this talk, I will describe the effect of this step on multiplicity modules and Külshammer-Puig classes. The Puig category, which carries more information than the fusion system of a block, is the main tool that is used to deliver these results. This is a joint work with Laurence Barker.