Kidney transplants across the blood-group barrier are medically feasible even if blood-group compatibility is preferred from a medical point of view. However, these types of transplants are motivated by the fact that they help in increasing the number of kidney transplants with living donors. This paper investigates priority matchings in a pairwise kidney exchange problem where blood-group incompatibilities may be present. As a priority matching not necessarily is unique, it is from a medical point of view natural to select a priority matching where the number of blood-group compatible exchanges is maximized among all priority matchings. The main result demonstrates that this can be achieved by solving an appropriately defined maximum weight matching problem.