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They belong to the Society of St Pius X, founded by a French archbishop, Marcel Lefebvre, in 1970 as a protest against the Second Vatican Council's reforms on religious freedom and pluralism.
BBC: Seminary sacks 'Holocaust bishop'
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He became a member of the Society of St Pius X, which had been founded by Archbishop Lefebvre in 1970 to counter the reforms in the Church made by the Second Vatican Council during the previous decade.
BBC: Profile: Bishop Richard Williamson
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The four bishops have been asked by the Vatican to recognise the authority of the Pope and the Second Vatican Council and talks are planned to seek to resolve the "open questions" in the Church's relationship with the Society of St Pius X.
BBC: Pope admits Holocaust row errors