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The evolution of fairness in dyadic relationships has been studied using ultimatum games. However, human fairness is not limited to two-person situations and universal egalitarianism among group members is widely observed. In this study, we investigated the evolution of favoritism and group fairness in a three-person ultimatum game (TUG) under a co-evolutionary framework with both strategy updating and partner switching dynamics. In the TUG, one proposer makes an offer to two responders and the proposal is accepted at the group level if at least one individual responder accepts the offer. Investigating fairness beyond dyadic relationships allows the possibility of favoritism because the proposer can secure acceptance at the group level by discriminating in favor of one responder. Our simulation showed that the proposer favors one responder with a similar type when the frequency of partner switching is low. In contrast, group fairness is observed when the frequency of partner switching is high. The correlation between strategy and neighborhood size suggested that partner switching influences the strategy through the proposer's offer rather than through the responder's acceptance threshold. In addition, the average degree negatively impacts the emergence of fairness unless the frequency of partner switching is high. Furthermore, a higher frequency of partner switching can support the evolution of fairness when the maximum number of games in one time step is restricted to smaller values.