What is the ruling concerning girls who have not reached puberty, and is it permissible for them to go out without a head covering, and is it permissible for them to pray without a head covering?

The guardian of women must discipline them with the manners of Islam, by enjoining them to cover themselves when going out for fear of temptation, and to accustom them to virtuous morals. He should also enjoin them to pray with a head covering (khimar), and even if she prays without it, her prayer is valid, based on the Prophet's (peace and blessings be upon him) saying: "Allah does not accept the prayer of a menstruating woman except with a khimar."