Quantum symmetric algebras (or noncommutative polynomial rings) arise in many
places in mathematics. In this article we find the multiplicative structure of
their Hochschild cohomology when the coefficients are in an arbitrary bimodule
algebra. When this bimodule algebra is a finite group extension (under a
diagonal action) of a quantum symmetric algebra, we give explicitly the graded
vector space structure. This yields a complete description of the Hochschild
cohomology ring of the corresponding skew group algebra.