Groups critical with respect to the spectra of alternating and sporadic groups

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 natural numbers, if $\omega$ is equal to the spectrum of $G$ and not equal to the spectrum of any proper section of $G$. We give full description of the finite groups critical with respect to the spectrum of the alternating group of degree 10 and the second Janko group.