  In this article, we study $G$-covers of klt varieties, where $G$ is a
reductive group. First, we exhibit an example of a klt singularity admitting a
$\mathbb{P}{\rm GL}_n(\mathbb{K})$-cover that is not of klt type. Then, we
restrict ourselves to $G$-quasi-torsors, a special class of $G$-covers that
behave like $G$-torsors outside closed subsets of codimension two. Given a
$G$-quasi-torsor $X\rightarrow Y$, where $G$ is a finite extension of a torus
$\mathbb{T}$, we show that $X$ is of klt type if and only if $Y$ is of klt
type. We prove a structural theorem for $\mathbb{T}$-quasi-torsors over normal
varieties in terms of Cox rings. As an application, we show that every sequence
of $\mathbb{T}$-quasi-torsors over a variety with klt type singularities is
eventually a sequence of $\mathbb{T}$-torsors. This is the torus version of a
result due to Greb-Kebekus-Peternell regarding finite quasi-torsors of
varieties with klt type singularities. On the contrary, we show that in any
dimension there exists a sequence of finite quasi-torsors and
$\mathbb{T}$-quasi-torsors over a klt type variety, such that infinitely many
of them are not torsors. We show that every variety with klt type singularities
is a quotient of a variety with canonical factorial singularities. We prove
that a variety with Zariski locally toric singularities is indeed the quotient
of a smooth variety by a solvable group. Finally, motivated by the work of
Stibitz, we study the optimal class of singularities for which the previous
results hold.
