Do You Have to Pray Zuhr after Jumu`ah?
Question

There is a town where there are nearly thirty-five mosques in which Jumu`ah prayers are held. When the worshippers finish Jumu`ah, they pray Zuhr afterwards. Is this action permitted or not?
Praise be to Allah.Do You Have to Pray Zuhr after Jumu`ah?
“It is a well-established fact in Islam (such that no Muslim has any excuse for not knowing), and the shar’i evidence indicates, that Allah has not prescribed any prayer at noon time on Fridays except one obligatory prayer for men who are residents and are settled (i.e., not traveling), free and accountable, which is Jumu`ah prayer .
If the Muslims offer Jumu`ah prayer, then there is no other obligation upon them, be it Zuhr or any other prayer . Jumu`ah prayer is the obligation of that time. The Prophet (peace and blessings of Allah be upon him), his Companions (may Allah be pleased with them), and the righteous salaf who came after them, did not pray any other obligatory prayer after Jumu`ah. 
The action to which you refer was introduced many centuries later, and undoubtedly it is one of the newly-invented innovations of which the Prophet (peace and blessings of Allah be upon him) said: “Beware of newly-invented matters, for every newly-invented matter is an innovation, and every innovation is a going astray.” (Narrated by Abu Dawud, 3991) 
Prophet Muhammad (peace and blessings of Allah be upon him) also said: “Whoever innovates something in this matter of ours [Islam] that is not part of it will have it rejected.” (Narrated by al-Bukhari and Muslim)
Undoubtedly praying Zuhr after Jumu`ah is something innovated which is not a part of the matter of the Prophet (peace and blessings of Allah be upon him), so it is to be rejected and is to be included in the innovations and misguidance against which the Prophet (peace and blessings of Allah be upon him) warned us. 
The scholars have drawn attention to that, including Shaykh Jamal al-Din al-Qasimi, in his book Islah al-Maaajid min al-Bida’ wa’l-‘Awaid and Shaykh Muhammad Ahmad ‘Abd al-Salam in his book al-Sunan wa’l-Mubtada’at. 
If someone were to say, “We only do that in order to be on the safe side, lest our Jumu`ah prayer was not valid,” our answer is that he should be told that the basic principle is that the Jumu`ah prayer is valid and is sound and that it is not obligatory to pray Zuhr, indeed it is not permissible to do so at the time of Jumu`ah for those for whom Jumu`ah is obligatory. 
Being on the safe side is prescribed in cases where the way of the Sunnah is not clear, and there is some doubt and uncertainty. But in a case such as this it is not the matter of doubt; rather we know from the evidence that what is obligatory is Jumu`ah prayer only, so it is not permissible to do something else instead or in addition to it on the grounds that this is an action with the intention of being on the safe side, or to institute a new way which Allah has not prescribed. 
Praying Zuhr at this time is contrary to the shar’i evidence which is well established and no Muslim has any excuse for not knowing, so we must not do this and we must beware of it. 
Source of waswas
There are no grounds for doing this, rather it comes from the waswas (insinuating whispers) of the Shaytan which he whispers to people so as to prevent them from following true guidance and institutes for them a religion which Allah has not allowed. 
This is similar to the way in which he makes the idea of “being on the safe side” when doing wudu attractive to them, until he makes wudu a torment for them, and it takes all their time; whenever they have almost finished, he whispers to them  that they have not done it properly or they have not done such and such. 
He does the same thing to some of them in the prayer; when a person says takbir for the prayer, he whispers to him that he did not say the takbir, and he keeps whispering to him that he did not say takbir, and the person keeps on saying takbir after takbir, until the first rak`ah is over, or until the recitation or most of it is over. 
This is one of the traps and snares of the Shaytan , who is keen to annul the good deeds of the Muslim and make him confused about his religion. We ask Allah to keep us and all the Muslims safe and sound from his traps, snares and insinuating whispers, for He is the All-Hearing, Ever-Near.
To conclude, praying Zuhr after Jumu`ah is a bid`ah and a going astray, instituting a new way which Allah has not prescribed. We must not do this; we must beware of it and warn people against it, and limit ourselves to doing Jumu`ah prayer only, as was the way of the Messenger of Allah (peace and blessings of Allah be upon him), his Companions after him and those who have followed them in truth until the present day. 
This is the truth concerning which there is no doubt. Imam Malik ibn Anas (may Allah have mercy on him) said: “The later generations of this ummah will not be reformed by anything except that by which the first generation was reformed.” This was also the view of other imams after him. And Allah is the source of strength.” (Majmu’ Fatawa Wa Maqalat Mutanawwi’ah Li Samahat Al-Shaykh Al-‘Allamah ‘Abd Al-‘Aziz Ibn ‘Abd-Allah Ibn Baz (May Allah Have Mercy On Him), Part 12, P. 363)
For more, please see this category: Friday Prayer.
And Allah knows best.