Is money earned from covering for colleagues for financial compensation permissible or impermissible, given that the chief physician is aware but the administration is not? And is purchasing these shifts considered assisting in sin and aggression?

Therefore, the head physician's permission for colleagues to cover shifts is not considered valid unless he is authorized by his employer, and it appears he is not. Hence, it is not permissible to cover colleagues' shifts due to the deception, falsification, violation of contract terms, dereliction of duty, and infringement upon patients' rights involved.

As for the money received from the employer for this work, you have no right to it because the physician is a private employee and is not allowed to appoint someone else without the permission of his employer.

However, if the absent physician were to pay the money from his own funds, it might be argued that it is permissible, despite the sin due to the violation, because the benefit itself is permissible.