Answer
Wa `alaykum As-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Brother, first of all, we would like to say that we are impressed by your question, which emanates from a thoughtful heart. May Allah Almighty help us all adhere to the principles of this true religion, Islam, and enable us to be among the dwellers of Paradise in the Hereafter, Ameen.
As for your question, we would like to state that there is nothing in the authentic sources of the Shari`ah to the effect that it is obligatory to mention the names of the four caliphs, the name of the Prophet’s uncle Al-`Abbas, the Prophet’s daughter Fatimah, and his grandsons in the second khutbah of Friday Prayer. However, it is preferable that the Friday
However, it is preferable that the Friday khutbah include praises of Allah, the Exalted, prayers upon the Prophet, admonitions, and Qur’anic recitations.
Also, there is nothing wrong in sending peace and blessings on the rightly-guided Caliphs and the Prophet’s family without believing that this is obligatory or one of the pillars of the khutbah without which it would be invalid.
Almighty Allah knows best.