Let $B$ be a normal affine $\boldsymbol{C}$-domain and let $A$ be a
$\boldsymbol{C}$-subalgebra of $B$ such that $B$ is a finite $A$-module. Let
$\delta$ be a locally nilpotent derivation on $A$. Then $\delta$ lifts uniquely
to the quotient field $L$ of $B$, which we denote by $\Delta$. We consider when
$\Delta$ is a locally nilpotent derivation of $B$. This is a classical subject
treated in [17, 19, 16]. We are interested in the case where $A$ is the
$G$-invariant subring of $B$ when a finite group $G$ acts on $B$. As a related
topic, we treat in the last section the finite coverings of log affine
pseudo-planes in terms of the liftings of the $\boldsymbol{A}^1$-fibrations
associated with locally nilpotent derivations.