  For $g\geq 2$, let $\mathrm{Mod}(S_g)$ be the mapping class group of closed
orientable surface $S_g$ of genus $g$. In this paper, we derive a finite
generating set for the liftable mapping class groups corresponding to
finite-sheeted regular branched cyclic covers of spheres. As an application, we
provide an algorithm to derive presentations of these liftable mapping class
groups, and the normalizers and centralizers of periodic mapping classes
corresponding to these covers. Furthermore, we determine the isomorphism
classes of the normalizers of irreducible periodic mapping classes in
$\mathrm{Mod}(S_g)$. Moreover, we derive presentations for the liftable mapping
class groups corresponding to covers induced by certain reducible periodic
mapping classes. Consequently, we derive a presentation for the centralizer and
normalizer of a reducible periodic mapping class in $\mathrm{Mod}(S_g)$ of the
highest order $2g+2$. As final applications of our results, we recover the
generating sets of the liftable mapping class groups of the hyperelliptic cover
obtained by Birman-Hilden and the balanced superelliptic cover obtained by
Ghaswala-Winarski.
