If you have a subgroup $\overline{M}$ of a quotient group $G/N$, the lift of $\overline{M}$ is a subgroup $M$ of $G$ such that the $\overline{M}$ is the image of $M$ under the projection homomorphism $G\rightarrow G/N$. (This is guaranteed to exist by the correspondence theorem.) So speaking of the lifted action of some group action implies that you are working with an element (or subgroup) of a factor group and you need to find the preimage of said element (or subgroup) in $G$.