On groups critical with respect to a set of natural numbers

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 $G$. For any natural number $n$, we construct $n$ finite critical groups with the same spectrum. We also give a complete description of finite groups critical with respect to the spectrum of the alternating group of degree 6 and the spectrum of the special linear group of dimension 3 over a field of order 3.