Christianity had already taken root in Britain but it only came under Catholic control with the arrival of the first official Christian missionary to England, sent by Pope Gregory the Great. Wikipedia tells us: He was a Benedictine monk who became the first Archbishop of Canterbury in the year 597. He is considered the "Apostle to the English" and a founder of the English Church. Died at Canterbury 604 or 605.
The Augustinians (Austin) friars are named for Saint Augustine of Hippo and so not connected to Augustine of Canterbury at all.