Is it permissible for a wife to set aside a portion of her allowance for her mother without her husband's knowledge, or to help her with money from her own work, and is his permission required for that?

If your mother is in need of financial support, it is your duty to provide for her as long as you are well-off, whereas her siblings are not obligated to do so. If she is not in need, you may spend on her from your own money without informing your husband, and he has no right to prevent you. If your husband gives you your maintenance allowance, you may give some of it to your mother, for a woman has the right to dispose of her maintenance as she wishes, as long as it does not harm her.