So either she did and TSA didn’t care, it she didn’t and they blundered.

I haven't found a story yet that says whether the firearm was in a checked bag or a carry-on and that really matters. If it was a carry-on then the TSA utterly failed. If it was checked baggage though then she likely followed the correct US process but took a gun to a country that didn't allow it and Australian customs caught it as they were doing the luggage scan.

Despite what you may think it IS possible to legally take firearms to other countries assuming you follow the process and the law at your destination. For example there's gobs of people from all over the world that fly into Colorado, Wyoming, and Alaska every fall with their hunting rifles. It's also possible, although difficult, to bring firearms into Australia.

Ultimately the TSA would have no definitive way to know whether or not she was actually allowed to bring the pistol to Australia, all they can reasonably do is make sure that it's stored safely and out of reach. The rest of it is on the person traveling.