With due respect to the previous posters, I don't think the question of why the chord would be referred to as a G#aug6 was really addressed. Let me have a go...
Consider the progression F/A to G. One nice way of scoring it would be as follows:
F -> G
C -> B
A -> G
That works well because no note moves by more than one step, and there's some contrary motion to add interest. But it could be more interesting. How about we add a bit of chromaticism:
F -> F# -> G
C -> C -> B
A -> Ab -> G
Chromaticism just means using notes which don't belong to the key. In this case we raised the F to an F# to create a smoother movement to the G above it, and lowered the A to an Ab in the direction of the G below it. In doing so we have created a new chord made up of the notes Ab, C and F#. Because the interval between the Ab and the F# is an augmented sixth, we call this, rather imaginatively, an augmented sixth chord.
This is how tritone substitution first came about, way back in the days before jazz. It wasn't thought of as tritone substitution, but as an altered 6-3 chord (a chord which features a sixth and a third above the root, i.e., as in this example, A-C-F, altered to Ab-C-F#). Anybody using this naming system presumably learned their theory the old way...
Over time, as composers pushed the boundaries of musical theory, this chord was used more and more often 'unprepared', for example without being preceded by the A-C-F chord in the previous example. It was also enhanced with the addition either an augmented fourth, or perfect fifth, i.e. D or Eb in the previous example.
By the time jazz theory came to be formulated, the old rules of music had been eroded to the point where it no longer seemed relevant to refer to a chord which contained the exact same notes as a seventh chord as anything but a seventh chord.
In conclusion: Ab7 for jazz, Abaug6 for classical music and, as has been pointed out previously, never G# anything.