Is it permissible to accept the increase in overtime pay granted by the manager for accomplishing the work, knowing that the recorded overtime hours do not match the fingerprint hours and are subject to review by the general manager?

If your superior is authorized by government bodies to encourage workers and disburse money to them in exchange for increased productivity, then it is permissible for you to take the salary, as it is considered a gift or a bonus. However, if he is not authorized, then you have no right to it, because he would be disposing of another's money without right. For "an agent does not possess the right to act beyond what is necessitated by his principal's permission." Therefore, clarify the matter with him.