Italian parents, presumably concerned over the pagan nature of modern-day Santa Claus, outlawed the practice of telling children that Santa delivered them their presents. Instead, the Vatican, unable to prove the existence of Santa but certain about the existence of witches, decided to tell kids that a kindly old witch, La Befana, delivers them. Rock solid, Vatican.
According to legend, the Wise Men asked La Befana to accompany them to see the infant Jesus. She refused, saying she was too busy. Now there's a lady who doesn't want anyone to get the wrong impression about her.