Is it permissible for a government employee to leave work with the permission of a superior who does not record his absence and considers him present, knowing that the superior records in the fingerprint report that he was on "official duty"? And is it permissible for him to leave for Friday prayer and return to work without official leave?

If the person in charge is authorized to grant permission for departure, then there is no blame upon you. Otherwise, what you receive from your salary for these times is unlawful for you. As for leaving for Friday prayer, it is a religious obligation and does not require permission. Ibn Qudamah said: "A monthly paid employee attends the Eids and Friday prayer, and this is not a condition."