There is an important difference between the Governor-General of Australia and the Governor General of Canada (apart from the hyphen in the title). The Constitution of the Commonwealth of Australia assigns specific responsibilities to the Governor-General, not to the Queen. Section
61 and
62 of the Constitution states that:
Executive power
The executive power of the Commonwealth is vested in the Queen and is exercisable by the Governor‑General as the Queen's representative, and extends to the execution and maintenance of this Constitution, and of the laws of the Commonwealth.
Federal Executive Council
There shall be a Federal Executive Council to advise the Governor‑General in the government of the Commonwealth, and the members of the Council shall be chosen and summoned by the Governor‑General and sworn as Executive Councillors, and shall hold office during his pleasure.
In theory, this means that even if the Queen moved permanently to Yarralumla, we would still need a Governor-General to keep the wheels of government turning. On the Queen's first visit to Australia it was planned that she would open the Commonwealth Parliament and attend a meeting of the Federal Executive Council. The plans were interrupted when it was realised that Sections 61 and 62 did not allow the Queen to do so. The
Royal Powers Act 1953 was quickly passed to enable the Queen to exercise her vested power.
I think there is also a difference between Australian State Governors and Canadian Provincial Lieutenant-Governors. If memory serves me right, Lieutenant-Governors are appointed by the Governor General, on the recommendation of the Provincial Premier. In Australia, however, since 1986 State Governors are appointed by the Queen on the advice of the State Premier. But State Premiers cannot advise the Queen on any other issue without prior agreement (e.g. if the Premier of South Australia advised the Queen to sack the Governor of Tasmania, the Queen is not bound to do so).
I have always liked the idea of a member of the Royal Family coming to Australia permanently as our King or Queen. I don't think it has been suggested as a serious option for many years now. I think it was in 1988, prior to a visit from the Duke and Duchess of York, that Ian Sinclair (National Party) said that the Duke should be invited to stay as King of Australia. Can you imagine what might have been? Queen Fergie!