  For gauge groups $U(1)$ and $SO(3)$ we classify invariant $G_2$-instantons
for homogeneous coclosed $G_2$-structures on Aloff-Wallach spaces $X_{k,l}$. As
a consequence, we give examples where $G_2$-instantons can be used to
distinguish between different strictly nearly parallel $G_2$-structures on the
same Aloff-Wallach space. In addition to this, we find that while certain
$G_2$-instantons exist for the strictly nearly parallel $G_2$-structure on
$X_{1,1}$, no such $G_2$-instantons exist for the tri-Sasakian one. As a
further consequence of the classification, we produce examples of some other
interesting phenomena, such as: irreducible $G_2$-instantons that, as the
structure varies, merge into the same reducible and obstructed one; and
$G_2$-instantons on nearly parallel $G_2$-manifolds that are not locally energy
minimizing.
