  The strong decays of the $5^1S_0$ $q\bar{q}$ states are evaluated in the
$^3P_0$ model with two types of space wave functions. Comparing the model
expectations with the experimental data for the $\pi(2360)$, $\eta(2320)$,
$X(2370)$, and $X(2500)$, we suggest that the $\pi(2360)$, $\eta(2320)$, and
$X(2500)$ can be assigned as the members of the $5^1S_0$ meson nonet, while the
$5^1S_0$ assignment for the $X(2370)$ is not favored by its width. The $5^1S_0$
kaon is predicted to have a mass of about 2418 MeV and a width of about 163 MeV
or 225 MeV.
