Mary,
What name is given for Edmund's mother? I cannot find a marriage for George Coleman in the NSW BMDs, so he may have brought his wife here with him from England. The 4th Regt was The 4th (The King's Own) Regiment of Foot and served in various parts of Australia from 1832 to 1837. Many of its members remained in Australia.
There is a list of names of those who remained (
www.regiments.org) which includes a William Coleman, but not George Coleman. That same website will give some info re the Regt, including its Museum, The King's Own Royal Regiment Mumeum, Lancaster. There are phone numbers and an email address - you could contact them for personal information re past members of the Regt.
The PRO, London will have military records, possibly even the Description Books. The Mitchell Library, Sydney, also will have records of any British regiment which served here. Description Books give the name, previous occupation, where from, age, physical description, etc of every soldier.
There are 2 deaths in Sydney you might consider for George, that is if you know he remained here. There is one in 1862 and one in 1872. If all other enquiries fail, you could try these as NSW death certificates show parents' names if known to the informant, place of birth, how long in the colony, etc. If you think he returned to England, the Museum records in Lancaster may be able to confirm this.
The website above also can tell you the names and dates of the ships which brought the 4th Regt to Australia.
Diana