If you want to read about the IRA in Australia (or more strictly, official attitudes to the IRA in Australia), there is an interesting post here on the Irish Diaspora Histories network by Evan Smith and Anastasia Dukova. It notes that the Australian security services had monitored Irish republicans there since the Easter Rising. But in the early 1970s, fear that conflict in Ireland could spill over to Australia saw an increase in focus on Irish republicans in Australia.
The post reminded me of a throw away comment about the IRA and Australia reported in a court case in 1939. Hugh McCluskey, Thomas Magill and Robert McCann were IRA volunteers from Belfast who were involved in the sabotage campaign in England which began in January 1939. On 18th February 1939, a police raid in Wheelys Road in Edgebaston in Birmingham found all three in a house with magnesium, metallic sodium, magnesium scrapings and detonator wire. These were basic bomb-making equipment as IRA volunteers had been trained to use them to manufacture detonation charges to use with ‘paxo’ explosives (made by mixing potassium chlorate and paraffin wax). Other incriminating items were also recovered from the house, but no guns.
McCluskey and McCann had both been imprisoned in Belfast previously for offences under the Civil Authorities (Special Powers) Act. In court, on 20th March, they received ten year sentences, while Magill received seven years. During the trial a detective claimed under oath that, when he was being arrested, McCluskey admitted to having been a member of the Belfast Battalion and said “It’s all right. We have not got a gun. If we had we should have put a bullet through all of you. You caught us napping. I am better off inside. I am shutting up, as one of our fellows went to Australia and they got him there.”
The press reporting, including many Australian newspapers such as the Sydney Morning Herald, highlight the reference to Australia. Unfortunately, none actually explore the implications of what McCloskey is reported to have said (any speculation is obviously predicated upon the police officer truthfully repeating something that McCluskey said).
There are a number of instances of the Belfast IRA, or individuals connected to the Belfast Battalion of the IRA, killing former members or others connected to the IRA in the 1920s or 1930s, such as Patrick Woods, Joe Hanna and Dan Turley. However, there doesn’t appear to be any reference to a former member being tracked to Australia and killed there over the same time frame. A review of the Belfast Brigade lists compiled in the 1930s identifies four from Belfast with addresses in Australia in the 1930s. These are George Fitzsimons (Engineering Battalion), Henry McCollum, Thomas Corry and John Myles (all A Company, 2nd Battalion). There doesn’t appear to be anyone of the same names who died in suspicious circumstances reported in the Australian press up to 1939. The RUC suspect list from the 1930s doesn’t note anyone as having emigrated to Australia during this time.
While it may be a reference to an IRA volunteer from somewhere other than Belfast, again, this doesn’t appear to be mentioned anywhere else that I have noted up to now. One possibility is that it is a garbled version of the story of James Carey, the informer whose evidence had led to the execution of five Invincibles in 1882. Carey was spotted on a ship while emigrating to South Africa and killed by another Invincible, Patrick O’Donnell, who was subsequently executed (O’Donnell was a great grand uncle of Patsy Dougan).

Anyone who knows of any cases where someone from Belfast died in suspicious circumstances prior to 1939 could add some details in the comments below.
You can read more on the Belfast Battalion of the IRA here.