What is the ruling on the imam omitting the supplication at the end of the Friday sermon and instead making it after rising from rukūʿ in the second rakʿah of the prayer?

Supplicating for believing men and women in the Friday sermon is a Sunnah according to the majority of jurists, and a pillar (rukn) according to the Shafi'is. As for singling out Friday for qunut in calamities or otherwise, it is not legislated. Imam Al-Shafi'i explicitly stated this, saying: "I do not know of anyone among them who narrated that he performed qunut therein [in the Friday prayer], unless it was included in the generality of his qunut in all prayers when he performed qunut for the killers of the people of Bi'r Ma'unah. There is no qunut in any of the prayers except Fajr, unless a calamity befalls, in which case the Imam may perform qunut in all prayers if he wishes."