Music Theory for the 21st-Century Classroom

It may be helpful to review the three types of augmented sixth chords before discussing how one voice leads them.

The salient accidental in all three augmented sixth chords is ♯$\hat{4}$, which almost always resolves upward to $\hat{5}$.

The ♯$\hat{4}$ scale degree also resolves to $\hat{5}$ when an augmented sixth chord resolves to the Cadential ${}_4^6$ chord.

With the enharmonic German augmented sixth chord (abbreviated ${\text{EnGer}}^{+6}$, containing ♭$\hat{6}$, $\hat{1}$, ♯$\hat{2}$, and ♯$\hat{4}$), ♯$\hat{2}$ and ♯$\hat{4}$ resolve upward to members of a major ${\text{I}}_4^6$ chord.

The exception to ♯$\hat{4}$ to $\hat{5}$ occurs when an augmented sixth chord resolves to ${\text{V}}^7$, in which case ♯$\hat{4}$ resolves to ♮$\hat{4}$, which is the 7th of the ${\text{V}}^7$ chord.

If you voice lead an augmented sixth chord to something other than $\text{V}$, ${\text{V}}^7$, or ${\text{i}}_4^6$, move all of the voices smoothly while avoiding objectionable parallels and employing proper doublings.