The Church of England (C of E) is the 'mother' church of the worldwide Anglican Communion.
(The term 'Anglican' was chosen as the name of the reformed Church that was opposed to Roman influence. It's an interesting choice because 'Anglican' is a Latin term.)
Roots of the C of E go back to the 6th century and their banner was traditionally the flag of England; the Cross of St. George.
The C of E owns thousands of church buildings in England and most of these happen to be the oldest buildings in the country.
In an attempt to shake the dust off this ancient image, the church adopted a modern logo in 1996: A simple Christian cross encompassed by overlapping 'c' (for 'Church') and 'e' (for 'England').
There is some unintentional similarity between the insignia of the C of E and the Crescent Cross.
And there's nothing wrong with that...
...Christ died for all mankind, whatever their religion.