Every fingering that fits a hand
100x
open
1003
open
1403
easy
1443
moderate
x887
easy
What is true of it
G♯aug is G♯, B♯ and D𝄪 — 1, 3 and #5 counted from G♯.
D𝄪 is a double sharp, and it has to be: every letter from G♯ upwards is used once, so the #5 cannot be written any other way.
The easiest of the 5 fingerings below is 100x: one finger, held at the 1st fret.
2 of them use open strings, which is what makes a shape quiet to change to.
1 string stays silent.
Where its notes are on the neck
The brass dots are G♯, the root. Everything else is a chord tone with its degree printed in it.
With a different note underneath
G♯aug/B♯
5047
G♯aug/D𝄪
x443
An inversion is the same chord with a different note lowest. The name changes to a slash name because the bass is what a listener hears as the foundation.