Fair Influence Maximization (FIM) seeks to mitigate disparities in influence across different groups and has recently garnered increasing attention. A widely adopted notion of fairness in FIM is the maximin constraint, which directly requires maximizing the utility (influenced ratio within a group) of the worst-off group. Despite its intuitive formulation, designing efficient algorithms with strong theoretical guarantees remains challenging, as the maximin objective does not satisfy submodularity, a key property for designing approximate algorithms in traditional influence maximization settings. In this paper, we address this challenge by proposing a two-step optimization framework consisting of Inner-group Maximization (IGM) and Across-group Maximization (AGM). We first prove that the influence spread within any individual group remains submodular, enabling effective optimization within groups. Based on this, IGM applies a greedy approach to pick high-quality seeds for each group. In the second step, AGM coordinates seed selection across groups by introducing two strategies: Uniform Selection (US) and Greedy Selection (GS). We prove that AGM-GS holds a $(1-1/e-\varepsilon)$ approximation to the optimal solution when groups are completely disconnected, while AGM-US guarantees a roughly $\frac{1}{m}(1-1/e-\varepsilon)$ lower bound regardless of the group structure, with $m$ denoting the number of groups.