What is the ruling on a woman fasting to make up for two days of Ramadan without her husband's knowledge, and is expiation required if her fast is impermissible?

A woman must make up the days she missed in Ramadan, even without her husband's knowledge. Her husband's permission is not required for a make-up fast. As for non-obligatory fasting, a woman should not fast when her husband is present except with his permission, due to the prohibition of the Prophet (peace and blessings be upon him) regarding that, "except for Ramadan."