What is the expiation for swearing falsely by Allah out of fear of damaging one's reputation and having things said about the swearer that are not true?

A false oath by Allah is one of the grave sins. It is called an "oath that immerses" (yamin ghamoos) because it immerses its taker in sin or in Hellfire. Its expiation, according to most scholars, is sincere repentance. Some scholars also add an expiation for an oath. If a person is forced to swear a false oath to protect their honor or wealth, it is permissible for them to do so, though it is better to use dissimulation (tawriya).