  We consider two different $\text{SU}(2)^2$-invariant cohomogeneity one
manifolds, one non-compact $M=\mathbb{R}^4 \times S^3$ and one compact $M=S^4
\times S^3$, and study the existence of coclosed $\text{SU}(2)^2$-invariant
$G_2$-structures constructed from half-flat $\text{SU}(3)$-structures. For
$\mathbb{R}^4 \times S^3$, we prove the existence of a family of coclosed (but
not necessarily torsion-free) $G_2$-structures which is given by three smooth
functions satisfying certain boundary conditions around the singular orbit and
a non-zero parameter. Moreover, any coclosed $G_2$-structure constructed from a
half-flat $\text{SU}(3)$-structure is in this family. For $S^4 \times S^3$, we
prove that there are no $\text{SU}(2)^2$-invariant coclosed $G_2$-structures
constructed from half-flat $\text{SU}(3)$-structures.
