  We consider the Brauer group ${\rm BM}'(k,G)$ of a group $G$ (finite or
infinite) over a commutative ring $k$ with identity. A split exact sequence
$$1\longrightarrow {\rm Br}'(k)\longrightarrow {\rm BM}'(k,G)\longrightarrow
{\rm Gal}(k,G) \longrightarrow 1$$ is obtained. This generalizes the
Fr\"ohlich-Wall exact sequence from the case of a field to the case of a
commutative ring, and generalizes the Picco-Platzeck exact sequence from the
finite case of $G$ to the infinite case of $G$. Here ${\rm Br}'(k)$ is the
Brauer-Taylor group of Azumaya algebras (not necessarily with unit). The method
developed in this paper might provide a key to computing the equivariant Brauer
group of an infinite quantum group.
