Is it permissible to return the greeting of someone who offers it to me during the Friday sermon?

Initiating the greeting (salām) during the Friday sermon is not legislated. As for returning the greeting, there is a difference of opinion: the Shāfiʿīs obligated it, and the Ḥanābilah permitted it, while the Mālikīs and Ḥanafīs consider it disliked (makrūh) or forbidden (ḥarām). The most preponderant view is prevention, because listening attentively to the sermon is obligatory, and speaking—even to return a greeting—distracts from this obligation, based on the Prophet’s (peace and blessings be upon him) saying: "If you say to your companion on Friday, 'Listen,' while the Imam is delivering the sermon, you have engaged in idle talk (laghā)." Therefore, one should only respond by gesturing. And whoever speaks while aware of this (prohibition) is sinful.