Nearly Kähler and Hermitian $f$-structures on homogeneous $\Phi$-spaces of order $k$ with special metrics

We consider arbitrary homogeneous $\Phi$-spaces of order $k\ge3$ of semisimple compact Lie groups $G$ in the case of a series of special metrics. We give formulas for the Nomizu function of the Levi-Civita connection of these metrics. Using these formulas and other relations for $\Phi$-spaces of order $k$, we prove necessary and sufficient conditions for the canonical $f$-structures on these spaces to lie in some generalized Hermitian geometry classes of $f$-structures: nearly Kähler ($NKf$-structures) and Hermitian ($Hf$-structures).