  Let $\text{Mod}(S_g)$ be the mapping class group of the closed orientable
surface of genus $g \geq 1$, and let $\text{LMod}_{p}(X)$ be the liftable
mapping class group associated with a finite-sheeted branched cover $p:S \to
X$, where $X$ is a hyperbolic surface. For $k \geq 2$, let $p_k: S_{k(g-1)+1}
\to S_g$ be the standard $k$-sheeted regular cyclic cover. In this paper, we
derive an explicit finite generating sets for $\text{LMod}_{p_k}(S_g)$ for $g
\geq 3$ and $k \geq 2$, and $\text{LMod}_{p_2}(S_2)$. For $g \geq 2$, as an
application of our main result, we also derive a generating set for
$\text{LMod}_{p_2}(S_g) \cap C_{\text{Mod}(S_g)}(\iota)$, where
$C_{\text{Mod}(S_g)}(\iota)$ is the centralizer of the hyperelliptic involution
$\iota \in \text{Mod}(S_g)$. Let $\mathcal{L}$ be the infinite ladder surface,
and let $q_g : \mathcal{L} \to S_g$ be the standard infinite-sheeted cover
induced by $\langle h^{g-1} \rangle$ where $h$ is the standard handle shift on
$\mathcal{L}$. As another application, we derive a finite generating set for
$\text{LMod}_{q_g}(S_g)$. Finally, we establish that for $k \geq 2$ and $g \geq
2$, $\text{LMod}_{p_k}(S_g)$ is self-normalizing in $\text{Mod}(S_g)$.
