Is it permissible to pay Zakat to help a sister cover the costs of a surgical operation, knowing that she is a widow and has a fixed income from her deceased husband?

It is permissible to give Zakat (definition) to a sister under two conditions: that her maintenance is not obligatory upon you (it becomes obligatory if you are wealthy and an heir to her), and that she does not possess the cost of the surgical operation. This is because a sick person who does not have the means for treatment is considered poor and eligible to receive Zakat. Giving her Zakat in such a case is better, as it is both charity (sadaqah (definition)) and an act of maintaining kinship ties (silat ar-rahim).