Spectral sequences for the cohomology rings of a smash product

Stefan and Guichardet have provided Lyndon–Hochschild–Serre type spectral sequences which converge to the Hochschild cohomology and Ext groups of a smash product. We show that these spectral sequences carry natural multiplicative structures, and that these multiplicative structures can be used to calculate the cup product on Hochschild cohomology and the Yoneda product on an Ext algebra.