On finite groups isospectral to the simple groups $S_4(q)$

The spectrum of a finite group is the set of its element orders. A finite group $G$ is critical with respect to a subset $\omega$ of the natural numbers if $\omega$ coincides with the spectrum of $G$ and does not coincide with the spectra of proper sections of $G$. We study the structure of groups with spectra equal to the spectra of the simple symplectic groups $PSp(4,q)$, where $q > 3$ and $q \neq 5$. In particular, we describe the structure of the groups critical with respect to the spectra of $PSp(4,q)$.