Is the husband obligated to provide an allowance for his wife who does not work outside the home, so that she can save money to help her widowed mother?

The husband is not obligated to spend on his wife's mother; rather, his responsibility is towards his wife and household. It is commendable for him to honor his wife's family and assist them in need from his charity (sadaqat) or zakat, especially if the wife's mother has no one else to support her. It is permissible for the wife to give her mother from the household's food and the like, provided it does not harm the household's expenses and she predominantly believes her husband would permit it. Both she and her husband will be rewarded for this, as the Prophet (peace be upon him) said: "When a woman spends from her house's food, without causing corruption, she will have her reward for what she spent, and her husband will have his reward for what he earned, and the keeper (of the house) will have the like of that. None of them will diminish the reward of the others in any way."