Influence Maximization (IM) has been extensively studied in network science, which attempts to find a subset of users to maximize the influence spread. A new variant of IM, Fair Influence Maximization (FIM), which primarily enhances the fair propagation of information, attracts increasing attention in academic. However, existing algorithms for FIM suffer from a trade-off between fairness and running time. Since it is a tough task to ensure that users are fairly influenced in terms of sensitive attributes, such as race or gender, while maintaining a high influence spread. To tackle this problem, in this paper, we propose an effective and efficient Community-based Evolutionary Algorithm for FIM (named CEA-FIM). In CEA-FIM, a community-based node selection strategy is proposed to identify potential nodes, which not only considers the size of the community but also the attributes of the nodes in the community. Subsequently, we design an evolutionary algorithm based on the proposed node selection strategy to hasten the search for the optimal solution, including the novel initialization, crossover and mutation strategies. We validate the proposed algorithm CEA-FIM by performing experiments on real-world and synthetic networks. The experimental results show that the proposed CEA-FIM achieves a better balance between effectiveness and efficiency, compared to the state-of-the-art baseline algorithms.