Is there an expiation upon the husband after his wife insisted on fasting a day in which she missed the Fajr prayer, and she had sworn to fast it, despite her suffering from anemia, exhaustion, breastfeeding, and childcare? Is it permissible for her to break her husband's oath upon her not to fast, and is it permissible for her to burden herself with what she cannot bear in this matter?

If you swore an oath upon your wife that she should not fast, and she did fast, then you owe expiation according to the majority of scholars, and this is the more cautious view. However, Ibn Taymiyyah held that this falls under the category of a laghw al-yameen (idle oath) for which there is no expiation.

It is not permissible for your wife to refuse your command to break her fast and to fast without your permission, even if she had vowed to fast, due to the Prophet's (peace and blessings be upon him) saying: "It is not lawful for a woman to fast while her husband is present except with his permission."

It is disliked for a sick person who would be harmed by the illness to fast an obligatory fast, and it is forbidden for him according to some scholars.