Is it permissible for a wife to save money from her husband's allocated household expenses to give to her needy relatives without his knowledge, and is there any doubt of it being haram money?

It is not permissible for a wife to spend from her husband's private money on herself beyond the usual maintenance, or to give anything from it to her family or others, except with his permission, due to the Prophet's (peace be upon him) saying: "No woman may spend anything from her husband's house except with her husband's permission." An exception to this is a small amount that is customarily overlooked, due to the Prophet's (peace be upon him) saying: "If a woman spends from her husband's food, without causing damage, she will have her reward for what she spent, and her husband will have his reward for what he earned, and the storekeeper will have the like thereof. None of their rewards will be diminished in any way." So, if what the woman saves for her family is small and her husband is pleased with it, then there is no harm; otherwise, she must seek his permission.