  Crowley and Nordstr\"{o}m introduced an invariant of $G_2$-structures on the
tangent bundle of a closed 7-manifold, taking values in the integers modulo 48.
Using the spectral description of this invariant due to Crowley, Goette and
Nordstr\"{o}m, we compute it for many of the closed torsion-free
$G_2$-manifolds defined by Joyce's generalized Kummer construction.
