An auxiliary chord is one which is inserted between two identical chords. Like the passing chord, it does not disrupt the “true” progression, but instead works as a decoration chord.
For example, chord V normally moves to chord I and not to chord IV. But an auxiliary chord IV can be inserted between two chord Vs, as a decorative chord: The important chords here are Va-Ia, on the strong first beat of each bar.
The auxiliary 6-4 is another progression where you can use a second inversion chord. The second inversion chord is inserted between two identical chords. The second inversion chord falls on a weak beat of the bar, just as it does in a passing 6/4.
Look at this auxiliary 6-4. The second inversion chord, Vc, falls on a weak (middle) beat, between two identical bass notes. It sits between two tonic root position chords.
In this example, the auxiliary 6/4 chord is iiic. It sits between two submediant first inversion chords.
Second inversion chords sometimes arise during a “pedal point”. A “pedal” is a note which is repeated or held for some time, while the chords around it change. Pedal notes are normally either the tonic or dominant notes.
Here is an example in Bach’s Prelude in C Major, from the first book of his “Well Tempered Klavier”. The extract is in C major, and the dominant pedal is the repeated G which can be seen in the bass line. The harmony changes above the pedal point. In bar 24, the chord is V7 in root position. In bar 25, it is C major in second inversion (boxed). This is a pedal 6/4 chord.
Pedal 6/4s can normally only be used within an ongoing pedal section; they are not normally used as the first or last chord of a pedal section, as they are too unstable sounding.