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Archbishop of canterbury
Elizabeth the first of England was crowned "Queen of England, Ireland and France".The Archbishop of Canterbury.
Yes
Whoever is the reigning Monarch of Britain.
Princess Anne sat with the Queen in the Royal Box. She stood in the front row between the Archbishop of Canterbury Rowan Williams and Vice-Admiral Sir Timothy Laurence.
The Prime Minister of Great Britain makes the appointment
Margaret was crowned Queen consort of England on 30 May 1445 at Westminster Abbey by John Stafford, Archbishop of Canterbury. She was the wife of King Henry VI.
The Coronation of Queen Victoria took place in Westminster Abbey, London on 28 June 1838. The service was conducted by the then Archbishop of Canterbury, William Howley.
The Archbishop of Canterbury is the spiritual leader and chief primate of the Church of England and the entire world-wide Anglican Communion. The incumbent is The Most Rev. Dr. Rowan Williams.
Archbishop of Canterbury, Primate of All England and Metropolitan
The first minister of Scotland, Alex Salmond, presented the one surviving copy of Mary, Queen of Scots' death warrant to the Blairs Museum, Aberdeen, Scotland. The warrant was obtained by The Archbishop of Canterbury's library at Lambeth Palace, London, for the sum of £71,485 (with the assistance of donations). Following talks between The Archbishop of Canterbury, Cardinal O'Brien (leader of the Catholic Church in Scotland) and the first minister, it was presented to Scotland.
Her Majesty Queen Elizabeth II is "Defender of the Faith" and Supreme Governor of the Church of England i.e., the Anglican Communion. She is the ecclesiastical superior of the Archbishop of Canterbury. This answer is from en.wikipedia.org under "Queen Elizabeth II".