Praise be to Allaah.
A woman has to make up the days that she did not fast in Ramadaan even if that is without her husband's knowledge, and when it comes to an obligatory fast, her husband's permission is not essential. So the fast of the woman mentioned is valid. As for non-obligatory fasts, a woman should not fast when her husband is present except with his permission, because the Prophet (peace and blessings of Allaah be upon him) forbade women to fast when their husbands are present without their permission, apart from Ramadaan.
And Allaah is the Source of strength. May Allaah send blessings and peace upon our Prophet Muhammad and his family and companions.
Fataawa al-Lajnah al-Daa'imah li'l-Buhooth al-Ilmiyyah wa'l-Ifta (10/353).