This was interesting since the Archbishop of Canterbury, while the leader of the Church of England, has not had any role, spiritual or practical, over the US church since around July 4, 1776. Confusion about the authority of the see of Canterbury may be traced to its earliest days, however.
I have to confess that I find Augie of Canterbury an uninteresting subject. I know, I know. He was the first Archbishop of Canterbury; a representative of the Roman, rather than Celtic Christian, tradition and was charged by Pope Gregory the Great to abide in peace with the other Christians in Britain who were already part of an established church. [Remember that Christianity came to the British Isles at the same time it was being preached in Rome, and was well established among the Celts a couple of centuries before the Roman emperor converted to Christianity.]
Instead, he insisted on imposing Roman rites on the Celts, a move which was not only contrary to Gregory's advice, but established the first fissure in the eventual breach between the Church of Rome and what would become, centuries later, the Church of England and, eventually still, The Episcopal Church of the United States. [Although, to be historically accurate, the Episcopal Church owes its identity not to the Church of England as much as it does the Scottish Episcopal Church, bishops of which consecrated the first bishop of the American church. It's from the Scots that we get our name, too.]
More can be read of him at this site.
[Above is a photo of the first parish I ever served as vicar/rector: St. Augustine of Canterbury in Edinboro, Pennsylvania. I'm not sure what goes on there these days, as they don't have a website!]