Does a mother sin if she asks a woman whom she knows to be a liar to swear on the Book of Allah that she is certain of her accusation against her of going to sorcerers?

The woman who accuses your mother of witchcraft is sinful and must repent. Your mother should remind her of Allah, that it is not permissible to accuse someone without proof. Your mother is not obliged to swear an oath to clear herself, but if she does, there is no harm in it. Your mother should not ask the other woman to swear an oath if she knows she will lie, unless there is a benefit in doing so, such as removing an accusation from your mother. Otherwise, she should not make her swear, as it aids Satan.