The augmented 6th chords came about (theoretically)before the Dominant 7th was recognized as being a chord. So basically the Itaian, German,& French chords are nothing but dominant 7th chords inverted. Here's an example of the Italian 6th (the voice leading here is very important)

Key of C

IV Chord(1st inv) Italian 6 V
F F# G
C C B
A Ab G

the Italian 6th is nothing more than a Ab7 minus the 5th. It was given the name augmented 6th becuase of the interval between Ab & F#