Question: My Australian friend says yes. I'm not so sure, considering the monarchy collects no taxes from England, much less a country in the commonwealth. What's the verdict? Can she say "you're fired" and Tony Abbot is out?

Answer: 
Section 61 of the Constitution says "The executive power of the Commonwealth is vested in the Queen and **is exercisable by the Governor-General** as the Queen’s representative [...]". 

Prime Minister Robert Menzies had to pass a Parliamentary standing order specially for Queen Elizabeth to be able to open Parliament when she visited Australia in 1954 because he received legal advice that, based on the above wording of the Constitution, that power was exercisable only by the Governor-General, not the Queen. So, for the Queen to open Parliament while she visited, a standing order had to be passed through Parliament: "[When Her Majesty the Queen is present in Australia and intends to indicate in person the cause of the calling together of Parliament, references in this chapter to the Governor-General shall be read as references to Her Majesty the Queen.](http://www.aph.gov.au/About_Parliament/Senate/Powers_practice_n_procedures/~/link.aspx?_id=FC08DF8337A843498C94A38FABA57B34&_z=z)" Note that this standing order relates only to the opening of Parliament, and is only effective when the Queen is present in Australia. 

Constitutionally, the Queen can't exercise the executive power required to dismiss the Prime Minister - only the Governor-General can do this. The Queen can instruct the Governor-General to dismiss a Prime Minister (and, it's worth noting that there's no constitutional or legal requirement for the Governor-General to follow the Queen's instructions), but she can't dismiss the Prime Minister herself.
1. The Queen Cannot sack the Prime Minister of Australia.
2. In 1975, (I was there) the Governor General did NOT sack the Prime Minister, he has no power to sack the Prime Minister.
3. What can happen is this. If the Parliament cannot get legislature through parliament, due to blockage of the other side. Again and again. And 'supply is blocked'... i.e. money to be allocated, Parliament comes to a standstill. The Prime Minister can approach the Governor General and request a 'Double Dissolution' Both the Upper and Lower houses are disbanded, and a new election for both upper and lower houses are called for. This is what occurred in 1975. Whitlam couldn't get legislature through. There was an election, and Malcolm Fraser and Liberals were voted in.