Is the Adhan required precisely on time, and is it permissible to make the Adhan after the designated prayer time, given that the time for the subsequent prayer has not yet begun?

The general rule is that Friday prayer is to be performed in villages and cities, not in the desert. If someone performs it in the desert, it is not valid for him, and he must pray Dhuhr. However, if the desert is close to a city or a settled village, then Friday prayer may be established there. If the desert is far, you are obligated to pray Dhuhr. If it is close, you are permitted to pray Friday prayer. There is no harm in performing the Adhan fifteen minutes or more or less after the time for prayer begins. It is not obligatory for the Adhan to be performed immediately upon the entry of the prayer time, though it is better not to delay it to inform the people. The entry of the prayer time can be known by the Shar'i (Islamic legal) signs.