These descendants of the priests ⌞couldn’t prove their families were Israelites⌟: the descendants of Hobaiah, Hakkoz, and Barzillai (who had married one of the daughters of Barzillai from Gilead and took that ⌞family⌟ name).
When Herod was king of Judea, there was a priest named Zechariah, who belonged to the division of priests named after Abijah. Zechariah’s wife Elizabeth was a descendant of Aaron.