Supererogatory Prayers Before and After Jumu’ah

In the name of Allah (The One). I send peace and blessings upon His Messenger (Al-Mustafa) and upon His Messenger’s family, companions, and those who follow him in righteousness until the Day of Judgement (Yowm Al-Qiyaamah).
Before Jumuah
There aren’t any narrations about praying Sunnahs before Jumu’ah. Shaykh Uthaymeen said,

“There aren’t any (perscribed) Sunnahs before it (Jumu’ah). So, you can pray whatever you want without intending a specific number. So you can pray 2 Rak’ahs or what ever you want, but when the Imam enters then you refrain.”

He continued to say: If someone was to say,

“Do you choose for me if I come to Jumu’ah that I busy myself with Salah or should I busy myself with reading the Qur’an?” The answer is that I see 2 Rak’ahs are a necessity, and they are Tahiyah Al-Masjid (greeting the Masjid). Other than that, the person has to look into what is best for him. If he is in a Masjid crowded with people and there are a lot of people moving around in front of you then obviously reading the Qur’an is more beneficial and will bring the soul more tranquility. And if you are in a place free from annoyances, then no doubt praying is better than reading. This is because Salah combines reading, remembrance (of Allah), Du’a, standing, sitting, bowing, and prostrating. Surely it is a garden of the gardens of worship, so it is better. Imam Ahmed was asked about an issue of knowledge and said, “Look at what is best for your heart and do that.”

After Jumuah
There are 3 different narrations dealing with this:

Abdullah ibn Umar narrated, “After praying (Salah) Al-Jumu’ah he ﷺ would pray two Rak’ahs in his house.”
[Bukhari and Muslim]
Abu Hurayrah narrated, “If one of you prays Jumu’ah then pray after it four (Rak’ahs).”
[Muslim]
Ibn Umar, “If he was in Makkah, then after Jumu’ah he would pray two Rak’ahs and then another four. And in Madinah he would pray two Rak’ahs in his house. He said, “The Messenger ﷺ used to do this.”
[Abu Dawud]

The scholars differ on what is to be done here and there are three different opinions:
The opinion of Shaykh Al-Islam Ibn Taymiyya who said,

“If you are going to pray the Sunnahs of Jumu’ah in the Masjid then pray two and if you are going to pray them at home then pray two also.”

There are different types, sometimes he ﷺ prayed four and sometimes he prayed two.
It is only four, because if there is a contradiction between the statement and the action of the Prophet ﷺ then you act upon his statement.
As for praying six Rak’ahs after Jumu’ah. Shaykh Uthaymeen said,

“This comes from a hadeeth of Ibn Umar where he said, “He ﷺ used to do this.” [Abu Dawud]. But that what comes in Bukhari and Muslim is that he used to pray four. It is possible to prove that the Prophet ﷺ used to pray two at home and commanded that we pray four after Jumu’ah, thus being six — four with his statement and two with his actions. That does make you ponder.”

And I send peace and salutations upon our Prophet Muhammad, his family and his companions.
[Source: Taken from Sharh Al-Mumt’i by Shaykh Ibn Al-Uthaymeen, volume 5 pages 76-78