On the existence of $G_2$ class structures on a strictly nearly Kähler six-dimensional manifold

The strictly nearly Kähler 6-manifold $(M, g, J, \omega)$ is researched. Since the class $G_2$ is the orthogonal complement to the class of nearly Kähler structures in the space of all classes of almost Hermitian structures, no strictly nearly Kähler structure can be simultaneously an almost Hermitian structure of the $G_2$ class. Can this class contain other structures, «close» to a strictly nearly Kähler structure, in the case of dimension six? There exist three families of almost Hermitian structures linked with the given structure $(g, J, \omega)$ on $M$, namely, $H_g$, $H_J$, and $H_\omega$ families of almost Hermitian structures with the same metric $g$, or the same almost complex structure $J$, or the same form $\omega$, respectively. The problem whether a structure of the $G_2$ class can be present among structures belonging to those families is studied. It is proved that $H_\omega$ and $H_J$ do not contain structures of the $G_2$ class. By an example of left-invariant structures on $S^3\times S^3=SU(2)\times SU(2)$, it is proved that this is nevertheless possible for structures from $H_g$.