In this paper, we introduced the notions of $\mathcal{L}\mathcal{A}$-lifting modules and $\mathcal{L}\mathcal{A}$-discrete modules, which are strong concepts of lifting modules and discrete modules, respectively. We provided some conditions of these modules and generalizations of $ss$-lifting modules and $ss$-discrete modules, respectively. Finally, we characterized $\mathcal{L}\mathcal{A}$-lifting modules via left perfect rings and strongly $\mathcal{L}\mathcal{A}$-discrete rings by using the notion of semiperfect rings.