No animal should be forced to be eliminated due to the human race trying to play god. These animals have a right to be on this earth just as much as we do. It is true that these feral felines have created a few extinctions of various species due to hunger and over population, but relocation, adoption, or some other action other than death could be a solution.
When considering the number of species these felines have caused to go extinct, it is nothing compared to what the human race has caused. In fact "Humans are killing off species thousands of times faster than nature creates them" according to a recent study by the WWF. Despite this heart aching fact, you dont see humans murdering other humans due to the extinctions we have created, and we are just as overpopulated as the feral cats of Australia. Poachers and other horrible people contribute to the loss of species each and every day, but of course we are humans, so an excuse can be made for us, right?
Here is the link to the feral cat situation
Here is the link to the WWF study