Abstract
Let ℑ be the class of all finite simple groups and f: ℑ → {Fitting classes of groups}. The Fitting class \( \mathfrak{F} \) = KR(f) = (G: OA,A′ (G) ∈ f(A) for any A ∈ 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.
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