  The aim of this paper is to investigate properties preserved and co-preserved
by coarsely $n$-to-1 functions, in particular by the quotient maps $X\to
X/\sim$ induced by a finite group $G$ acting by isometries on a metric space
$X$. The coarse properties we are mainly interested in are related to
asymptotic dimension and its generalizations: having finite asymptotic
dimension, asymptotic Property C, straight finite decomposition complexity,
countable asymptotic dimension, and metric sparsification property. We provide
an alternative description of asymptotic Property C and we prove that the class
of spaces with straight finite decomposition complexity coincides with the
class of spaces of countable asymptotic dimension.
