The “Neapolitan 6th” chord is a 1st inversion of the supertonic [chord two] triad but with the root and fifth lowered a half step. 
In modern terms we could think of it as a flattened second chord 1st inversion. Below is a more popular common approach.
Minor Key example:
[In this case only the root is lowered as the C minor key signature already has the Ab in the key signature]
OTHER COMMON AUGMENTED 6 TH CHORDS
These chords can occur on the flattened submediant [flattened chord 6] of the major scale or the normal submediant [chord 6] of the minor scale.
FRENCH SIXTH
ITALIAN 6TH
GERMAN 6TH
9ths 11ths and 13ths harmony:
One last idea using the 7th chord as a Diminished chord followed by the dominant 7th 1st inversion to C the Tonic chord.
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