Is it permissible to combine the Dhuhr and Asr prayers after the time for Dhuhr has ended and before the time for Asr has begun, due to returning from school, or is it obligatory to pray only Dhuhr as a shortened prayer, or is there another solution?

Delaying prayer beyond its prescribed time is a major sin, as Allah says: "So woe to those who pray, [but] who are heedless of their prayer" (Al-Ma'un: 4-5), and His saying: "Then succeeded them later generations who neglected prayers and pursued passions; so they are going to meet evil" (Maryam: 59). Being preoccupied with studies is not an excuse for delaying prayer or combining them. One must repent and perform the prayer on time, taking care to be ritually pure and to pray in any pure place. If this is difficult and performing the prayer [on time] would cause harm, then it is permissible to combine Dhuhr and Asr prayers out of necessity, but each prayer should be performed as four units (rak'ahs).