Is a wife considered divorced twice after swearing falsely on the Qur'an, and must she inform her husband of her lie, and is it permissible for her to have intercourse with him with the intention of returning to his marital bond?

If a reliable authority issues a fatwa that the first divorce did not occur, then that pronouncement of divorce is not counted. As for the husband making the divorce conditional on his wife lying about what she swore upon, then upon the occurrence of the lie, one divorce takes place. Repentance from the forbidden lie is obligatory. If the husband has intercourse with his wife after the divorce has occurred and before he becomes aware of it, then the intercourse is considered a رجعة (revocation of divorce) and is permissible for the wife during her 'iddah (waiting period), even if the husband was unaware of the divorce having occurred, because it is lawful for her. She must inform her husband that the condition upon which the divorce was suspended has occurred.