What is the expiation for swearing a false oath upon the Qur'an due to fear of divorce and the husband's abusive behavior, and how does Allah forgive this sin?

Pure lying is impermissible, and swearing an oath upon it is a Ghamus oath, which plunges its maker into sin or Hellfire. The sin is compounded if the oath is sworn upon the Mushaf. However, lying and swearing an oath upon it are permissible if there is a need for it, to achieve a benefit or avert harm, provided that equivocation (Tawriyah) is used as much as possible. If your oath upon the Mushaf falls into this category, then there is no blame upon you. Nonetheless, it is more cautious to offer expiation for an oath (kaffarat yamin) to resolve the difference of opinion among scholars regarding the necessity of expiation for a Ghamus oath.