  A product of a K3 surface $S$ and a flat 3-dimensional torus $T^3$ is a
manifold with holonomy $SU(2)$. Since $SU(2)$ is a subgroup of $G_2$, $S\times
T^3$ carries a torsion-free $G_2$-structure. We assume that $S$ admits an
action of $\mathbb{Z}^2_2$ with certain properties. There are several
possibilities to extend this action to $S\times T^3$. A recent result of Joyce
and Karigiannis allows us to resolve the singularities of $(S\times
T^3)/\mathbb{Z}^2_2$ such that we obtain smooth $G_2$-manifolds. We classify
the quotients $(S\times T^3)/\mathbb{Z}^2_2$ under certain restrictions and
compute the Betti numbers of the corresponding $G_2$-manifolds. Moreover, we
study a class of quotients by a non-abelian group. Several of our examples have
new values of $(b^2,b^3)$.
