Is it permissible for a sister to ask her friend to call her after the caller swore she would not call her friend until the friend called her first?

If your intention by your oath was not to be the first to initiate contact with your friend, then you break your oath if you initiate a call to her, especially if you instructed your sister to call her from your phone or hers, because the basis of an oath is the intention of the one who swears it.

However, if that was not your intention, then you do not break your oath by calling her after your sister has called her. Ibn Qudamah said: "Oaths are referred back to the intention."

If there is a benefit in continuing your relationship with your friend and no harm results from it, then it is better to expiate your oath and maintain contact with her, based on the saying of the Prophet, peace and blessings be upon him: "Whoever swears an oath and then sees something better than it, let him do that which is better and expiate his oath."

The expiation for an oath is feeding ten needy persons, or clothing them, or freeing a slave. Whoever cannot find any of these, then fasting for three days.