Is it permissible for the Imam to pray the Dhuhr prayer four rak'ahs after the Friday prayer?

What this Imam is doing has no basis [in Islamic law], nor should it be acted upon. Even though it is the view of the Shafi'i school regarding the multiplicity of Jumu'ah (Friday) prayers in one city, when the condition for multiple Jumu'ahs is not met. In such a case, they (the Shafi'is) obligate the latecomer to pray Dhuhr (noon prayer). If there is doubt about who started first, all are obligated to repeat their prayer. If it is known that they all began the takbirat al-ihram (opening takbir) simultaneously, then everyone must repeat the Jumu'ah prayer if there is still time. Whoever repeats it first, their Jumu'ah is valid, and the others must pray Dhuhr. There is no doubt that this has no basis in the Qur'an, Sunnah, Ijma' (consensus), or sound Qiyas (analogical reasoning).