Is my mother obligated to offer an expiatory oath (kaffarat yamin) by fasting three days, after she swore by Allah to pay for her nephew's 'Umrah, and then she found out that her brother had paid the amount on her behalf without her knowledge?

If the mother swore that she would return the money so that her son would not have a favor over her, and her son paid the money on her behalf, then no expiation is required of her. However, if her intention was to pay the money herself, then she must give the money to her son if he paid it on her behalf, or an expiation for breaking an oath if he has not yet paid it. This expiation is feeding ten poor persons or clothing them. If she is unable to do so, then it is fasting for three days.