  This article is motivated by the following local-to-global question: is every
variety with tame quotient singularities globally the quotient of a smooth
variety by a finite group? We show that this question has a positive answer for
all quasi-projective varieties which are expressible as a quotient of a smooth
variety by a split torus (e.g. simplicial toric varieties). Although simplicial
toric varieties are rarely toric quotients of smooth varieties by finite
groups, we give an explicit procedure for constructing the quotient structure
using toric techniques.
  This result follow from a characterization of varieties which are expressible
as the quotient of a smooth variety by a split torus. As an additional
application of this characterization, we show that a variety with abelian
quotient singularities may fail to be a quotient of a smooth variety by a
finite abelian group. Concretely, we show that $\mathbb{P}^2/A_5$ is not
expressible as a quotient of a smooth variety by a finite abelian group.
