Possibly Ernest Austin Cooke who was a teacher at the time of his enlistment. His next of kin was his father who lived in Crawley, England. After the war, he enrolled and completed a BA at Sydney University and taught at Sydney Grammar School.  He was ordained as a Catholic priest in Rome in 1932 and returned to the UK in the 1930s. After the Second World War he returned to Australia. See his obituary The Catholic Weekly, 14 December 1950, p5, and Beyond 1914-The University of Sydney and the Great War.  A connection with Mosman Preparatory School could not be found. 

A definite connection with Mosman not confirmed but there is an Ernest Cooke at 42 Raglan Street, Mosman (Sands Directory, 1914).  [JSB]