The GST applies on the final price at the pump. That's a fact. That doesn't mean that all of the fuel charge will bear GST. There are situations where some of that may be reflected in the price of groceries, for instance. Groceries are a zero-rated product for GST purposes.
It will also depend on the behaviour of individuals. They may pay more at the pump, including some additional GST, but if there's a substitution that is happening, and to pay for that they reduce somewhat other discretionary expenses on which they would have paid GST, from looking at the application of a fuel charge, then it's a wash.
For the federal government, that doesn't result in additional GST, per se.