What should we do, as we have falsely sworn upon the Qur’an out of fear of the anger of our father, who suffers from chronic illnesses, after he thought we were with our brother who was expelled from the house?

Swearing falsely is a grave sin, and it is even more severe if it is reinforced by bringing a copy of the Qur’an. The majority of scholars hold that whoever swears a false oath intentionally does not owe expiation; rather, he must repent, because it is a sin too great to be effaced by an oath’s expiation. Whereas, Imam Al-Shafi’i and a group of scholars believe expiation is obligatory, based on the generality of Allah’s statement: “Allah will not impose blame upon you for what is unintentional in your oaths, but He will impose blame upon you for what you intended in your oaths.” [Al-Ma’idah: 89]. Combining repentance with expiation is safer. As for swearing falsely, it is permissible if it is necessary to attain a right or avert harm that cannot be achieved otherwise, as in the Hadith of Umm Kulthum, may Allah be pleased with her: “He is not a liar who reconciles between people and conveys good or says good.” In your case, fearing for your sick father that his crisis might worsen, we do not see a sin upon you for the oath, but it would have been better to use dissimulation (tawriyah). You should offer expiation as a precaution.