A G♯ Major 6 (G♯6) chord contains the notes G#, D#, C, F. It is built from the major 6th formula 1 - 3 - 5 - 6, with G# as the root.
Should I write this chord as G♯6 or A♭6?
Follow the key. Music in A♭ major, D♭ major or F minor writes A♭6, because those keys are spelled with flats and a G♯ would look out of place on the stave. Music travelling through E major or B major writes G♯6. The fretboard shape is identical, so the choice is about readability, not sound.