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→Augmented
Augmented chords consist of root, major third, and augmented 5th. An augmented fifth is equivalent to two major thirds. An augmented fifth + a major third = an octave. Therefore, the distance between root and 3rd, 3rd and 5th, and 5th and octave are all the same - a major third.
There are no real 7th chords - augmented chords tend to be associated with the whole-tone scale, which only has 6, not 7 notes. In the 7-note scale, you have 1st (C), 3rd (E), 5th (G), 7th (B), 9th (D), 11th (F), and 13th (A) before you reach the 15th (C), which is 2 octaves above the 1st. In the 6-note scale, you have 1st (C), 3rd (E), 5th (G#), 7th (C). The 7th note is the octave, and the chord simply repeats.
There aren't distinct inversion shapes - the first inversion of C aug (C E G#) is (E G# C), which is equivalent to (E G# B#) or E aug. So the inversions are identical to using a different root note. In other words, the inversions' shapes are identical to the root position shapes below.