An analog of Shemetkov's conjecture for Fischer classes of finite groups

We describe some methods for constructing Fischer classes of finite groups by means of the operators defined by given properties of Hall $\pi$-subgroups. It is in particular proved that, for a Fischer class $\mathfrak F$ and a set of primes $\pi$, the class of all finite $\pi$-soluble $C_\pi\mathfrak F$-groups, i.e., of all groups whose Hall $\pi$-subgroups belong to $\mathfrak F$, is a Fischer class.