Is it permissible to take a sister’s share of the inheritance, registered in the questioner’s name, after she betrayed the trust and seized the questioner’s and his mother’s money?

If it is proven that your sister wrongfully took money and severed ties of kinship, she must repent and restore the injustices to their rightful owners. Repentance for wrongs against people includes returning the money or seeking the owner's forgiveness.

If you are unable to recover your right from her, you may take it from her share of the inheritance that is under your control. This is known as "seizing one's right" (al-Dhafr bil-Haqq), provided that you do not take more than your right, that you are safe from scandal and punishment, and that you cannot obtain your right through the judiciary.