What is the ruling on praying four rak'ahs after the iqamah in the mosques of Northern Cyprus, as well as their doing so in Friday prayers after the sermon and the adhan, along with the speed of their bowing and prostrating, and is it obligatory to pray with them in congregation?

The answer summarizes the ruling on praying four rak'ahs in the mosque on Friday in several scenarios:

1. Praying four rak'ahs after the adhan and before the iqamah for Dhuhr prayer: This is desirable and legislated before Dhuhr, representing the Sunnah (pre-Dhuhr) prayer. 2. Praying four rak'ahs after the second adhan (for Jumu'ah) and when the khatib (preacher) has begun the sermon: This is unanimously considered makruh tahrim (prohibitively disliked), as voluntary prayers are forbidden for those inside the mosque during the sermon, as it conflicts with the obligation to listen to the khatib. 3. Praying four rak'ahs immediately after the Jumu'ah prayer (the Dhuhr prayer): This is an innovation (bid'ah), misguidance, and a newly invented practice for which Allah has not given permission. Verified imams have criticized those who advocate this practice—such as some Hanafi and Shafii jurists who permitted it as a precaution for fear that the Jumu'ah prayer might not be valid—clarifying its lack of legitimacy. They assert that the Jumu'ah prayer is the only obligatory prayer at that time, and sufficing with it is the truth upon which the Prophet (peace be upon him) and his companions proceeded.

Regarding abandoning Jumu'ah due to this innovation, that is not permissible. One must pray Jumu'ah behind the imam, even if he is an innovator or a transgressor, as long as the innovation is practical and does not affect the validity of the prayer itself. Then, the worshipper departs and does not endorse the innovation.