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Peter, Princess Bertha came as a Christian to Kent marrying Prince Ethelbert. Her father was a Frankish king.
Prince Ethelbert later succeeded his father King Eormenric and became King of Kent, and his Consort converted him to the Faith.
Not Augustine.
Augustine came later, because the pope had sent him to Kent which already had a Christian Monarch (King Ethelbert), with the mission to establish a bishop's seat in London and one in York (therefore not Canterbury), and to impose papal supremacy over the Celtic Churches in Cornwall, Wales and Scotland.
Augustine failed in all his missions, as the Celtic bishops refused to buckle down to his arrogance (see the Venerable Bede), and because neither the Anglo-Saxons of London nor those of York would convert to Augustine's catholicism.
They remained pagans.
King Ethelbert was already Christian, and had already established the English Church in Canterbury, where he had dedicated a preexisting Celtic church to St. Martin of Tours for his Consort Queen Bertha to practice her Christian Faith.
King Ethelbert simply allowed Augustine to remain in Canterbury and to be bishop there, as he had nowhere else to go.