Every fingering that fits a hand
322x
easy
3221
moderate
3225
moderate
3665
hard
xaa9
easy
What is true of it
A♯aug is A♯, C𝄪 and E𝄪 — 1, 3 and #5 counted from A♯.
C𝄪 and E𝄪 are double sharps, and they have to be: every letter from A♯ upwards is used once, so the 3 cannot be written any other way.
The easiest of the 5 fingerings below is 322x: 3 fingers, held at the 2nd fret.
None of them uses an open string, so this chord moves up the neck unchanged — the same shape 1 fret higher is the chord a semitone above.
1 string stays silent.
Where its notes are on the neck
The brass dots are A♯, the root. Everything else is a chord tone with its degree printed in it.
With a different note underneath
A♯aug/C𝄪
x221
A♯aug/E𝄪
x665
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.