Demented 6th Chords

I think this is right: If you find yourself on a IV chord, Option one: put it in the 1st inversion (b) and play it. In C the IV chord would be F major, in first inversion the bottom of the triad is A, the middle note is C, and the top note is F. Then you get to make an augmented sixth! Just lower the bottom note of that triad (this would be an A going to A-flat) and raise the top note (F to F#.) Now you have a triad that's Ab, C, and F#. This is the Italian augmented sixth. It resolves (by step) to a I or a IV chord. Alternatively, do all the above steps and add a flat seven to the chord (this would be an Eb.) That is the German augmented sixth. It normally resolves to a I because it's hard to avoid parallel fifths resolving to a V (because that Eb makes it hard.) Option two: there's the French aug 6. That's on a ii chord. The ii chord (say, Dm) is put in the 2nd inversion (c) such that the triad is A, D, and F. Lower the bottom (A becomes Ab), raise the top (F becomes F#) and add in a flat 7 (this is a C because we're flatting the seven of the ii chord) and resolve to the I or V.