It is said that a wife requires the permission of her husband for Nafl fasting.

Answer

If the husband is not at home, the wife does not require his permission to keep Nafl fasts. A husband who is absent does not have the right to prevent his wife from Nafl fasting. Only if he is present should the wife seek permission. She should simply inform him that it is her intention to fast 'tomorrow'. If he does not object, then it will signify his permission.