Critical Non-Singly-Generated Totally Canonical Fitting Classes of Finite Groups

Let $\mathfrak I$ be the class of all finite simple groups and $f\colon\mathfrak I\to\{\text{Fitting classes of groups}\}$. The Fitting class $\mathfrak F=KR(f)=(G:O^{A,A'}(G)\in f(A)\text{ for any }A \in K(G))$ is referred to as a canonical Fitting class with satellite $f$. A non-singly-generated totally canonical Fitting class $\mathfrak F$ is said to be critical non-singly-generated if all proper totally canonical Fitting subclasses in $\mathfrak F$ are singly generated. In the paper, a complete description of the structure of the critical non-singly-generated totally canonical Fitting classes of finite groups is obtained.