The augmented 7th chord are the extended version of the aug chord. Notice that aug7 chords are also referred to as 7#5 chords.
Compared to the aug triad, the aug seventh include a flattened 7th. To summarize the aug7 chord from a theoretical point of view, the chord consists of a root, a major third, an augmented fifth and a flattened seventh. If Cmaj7 and Caug7 is compared, what happens is that C, E, G, B changes to C, E, G#, Bb.
These chords don't occur frequently, but can add subtle harmonic texture. For examples of aug7 in action, see suggestion of progressions below.
Aug7 chords open and closed positions
The diagrams above show possibilities for open-position augmented chords.
The diagram version of Aug7 omits the 5th (G). An option that include the 5h is X03223. Baug7 can also be played with an omitted 5th (A) as X21003.
It's very hard to find a fingerings that could be handled for Caug7 in open position. Notice the relationship with the Caug7 diagram version and the Cmaj7 fingering 8X998X.
A peculiar circumstance is that Daug7 also can be played with the same shape five frets up with another tone order (XX0576).
The Faug7 and Gaug7 fingerings present one of the possible movable shapes for this chord type. One way to play Gaug7 as an open chord is XX5441.
Aug7th chords with flats and sharps
Since these are not played all that often, they are only presented in shortened notation:
C# / Db aug7: X43405
D# / Eb aug7: X6542X
F# / Gb aug7: 2X233X
G# / Ab aug7: 4X455X
A# / Bb aug7: XX4332
You only need to learn one movable shape to play all aug7 chords:
The shape above has the root on the sixth (i.e. the thickest). So to play, for example Abaug7, the bass note should be on the fourth fret.
An alternative shape with a inversion:
Aug7 shape 2
For example, Gbaug7/E would be played with the bass note on the fourth fret.
Chord progressions with aug7 chord
Aug seventh chords are often used directly after a major chord with the same root note, here are some examples:
E - Eaug7 - Amaj7
D6 - Daug7 - G6
Cmaj7 - Caug7 - Fmaj7
G6 - Gaug7 - Cmaj7
The following progression using an ascending harmony on the fourth string.
A - Aaug7 - F#m7
In a jazz-blues context:
E7 - Baug7 - B7
Some of the progression has similarities as you may notice.