Federated contextual bandits (FCB), as a pivotal instance of combining federated learning (FL) and sequential decision-making, have received growing interest in recent years. However, existing FCB designs often adopt FL protocols tailored for specific settings, deviating from the canonical FL framework. Such disconnections not only prohibit these designs from flexibly leveraging canonical FL algorithmic approaches but also set considerable barriers for FCB to incorporate growing studies on FL attributes such as robustness and privacy. To promote a closer relationship between FL and FCB, we propose a novel FCB design, FedIGW, which can flexibly incorporate both existing and future FL protocols and thus is capable of harnessing the full spectrum of FL advances.