In this paper, we consider the minimax fractional programming Problem (FP) in which the functions are locally Lipschitz $(G,\beta)$-invex. With the help of a useful auxiliary minimax programming problem, we obtain not only $G$-sufficient but also $G$-necessary optimality conditions theorems for the Problem (FP). With $G$-necessary optimality conditions and $(G,\beta)$-invexity in the hand, we further construct dual Problem (D) for the primal one (FP) and prove duality results between Problems (FP) and (D). These results extend several known results to a wider class of programs.