Some men from the companions of the Prophet ﷺ reported regarding the funeral prayer that the imam should pronounce the takbir, then send blessings upon the Prophet ﷺ, and devote the supplication sincerely to the deceased during the three takbirs, then give the salam quietly as he concludes. The sunnah is that those behind him do the same as the imam.
أخبره رجال من أصحاب النبي ﷺ في الصلاة على الجنازة أن يكبر الإمام، ثم يصلي على النبي ﷺ، ويخلص الصلاة- في التكبيرات الثلاث، ثم يسلم تسليما خفيا حين ينصرف، والسنة أن يفعل من وراءه مثل ما فعل الإمام.
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Recorded by Hakim (1/360), and from him Bayhaqi (4/39–40), from Ismail bin Ahmad the merchant, who said, Muhammad bin Hasan Asqalani narrated to us, Harmala bin Yahya narrated to us, Ibn Wahb narrated to us, Yunus informed me, from Ibn Shihab, who said, Abu Umama bin Sahl bin Hunayf informed me, and he was among the chiefs and scholars of the Ansar, and the sons of those who were present at Badr with the Messenger of Allah ﷺ informed him, that men from the companions of the Messenger of Allah ﷺ spoke about the prayer over the funeral, and he related it.Hakim said, Zuhri said, "Abu Umama narrated that to me while Ibn Musayab was listening and did not object to it." Ibn Shihab said, "I then related what Abu Umama had informed me of the sunnah regarding the prayer over the deceased to Muhammad bin Suwayd, who said, 'And I heard Dahhak bin Qays narrating from Habib bin Maslama about a prayer he performed over the deceased similar to what Abu Umama had told us.'"Hakim said, "This is a sahih hadith according to the standard of the two Shaykhs, though they did not record it, and there is nothing more sahih than it regarding a single tasleem over the funeral prayer." End of quote.It was also recorded by Ibn Jarud in Muntaqa (540), from Muhammad bin Yahya, who said, Abdul-Razzaq narrated to us, who said, Mamar informed us, from Zuhri, who said, I heard Abu Umama bin Sahl bin Hunayf narrating to Ibn Musayab, saying, "The sunnah regarding the prayer over the funeral is that you say the takbir, then recite Umm Quran, then send blessings upon the Prophet ﷺ, then devote the supplication entirely to the deceased, and you do not recite except in the first takbir, then you say the tasleem quietly to yourself, turning to your right."This is a sahih chain of transmission. Hafiz said in Talkhis (2/122), after relating it from Ibn Jarud, "The narrators of this chain are recorded in the Sahihayn." I say, its apparent form is mursal, but it has been established as connected as shown previously.Hakim then mentioned the hadith of Abu Hurairah that follows as a supporting witness for the hadith of Abu Umama. His statement, "quietly to himself," means that he does not raise his voice sharply.