All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
The Prophet did not use to pray any prayer before the Friday prayer and after its Athaan (call to the prayer), nor did his companions use to do so. Imam Ibn Taymiyyah said: “As for the Prophet he did not use to pray any prayer between the Friday prayer and its Athaan and nobody reported this from him. In the era of the Prophet the Athaan used to be called only after the Prophet sat on the Mimbar (pulpit). Only then, would Bilaal call the Athaan. Then, the Prophet delivers the two Friday sermons. After that, Bilaal would call the Iqaamah before the Prophet leads the prayer. So, there was no possibility that either the Prophet or any of the Muslims praying with him, to pray any prayer between the Athaan and the Friday prayer.
Ibn Al Qayyim said: “Anyone who believes that they (the Prophet and the companions) used to pray any sort of prayer after Bilaal called the Athaan is ignorant of the Sunnah.”
In fact, they did not use to pray any prayer in that time. This is also the opinion of Maalik, and the most famous opinion of Ahmad; it is also one of the two opinions narrated from Ash-Shaafi’i…”
We conclude, on the basis of the above evidence, that the most preponderant opinion of the scholars over this issue is that 'the Muslim should not pray two Raka'hs after the Friday Athaan.’ So, what some people call 'The Sunnah of Friday' is not proved by sound evidence.
Moreover, what some Muath-thins (the persons calling the Athaan) say viz.: 'Pray the Friday Sunnah may Allaah have mercy on me and you', is an innovation which has nothing to do with the true religion. But we advise people praying with them to teach these Muath-thins in a kind way what should be done and to beware of anything that could cause disunity and disintegration. However, this is an innovation and one should not do it himself.
The Sunnah is to pray four Raka'hs after the Friday prayer. Abu Hurayrah narrated that the Prophet said: “When you pray Friday prayer, pray four Raka'hs after it.” [Muslim and others]
But if one prays two Raka'hs only, then there is no harm in that. Naafi' reported from Ibn ‘Umar that he used to leave the mosque after performing the Friday prayer and used to pray two Raka'hs in his home. Then he said that the Prophet used to do that.
For more benefit on the Sunnah prayers for the Friday prayer, please refer to Fataawa 90213, 17270, and 86758.
Allah knows best.