Finite groups with small unbalancing 2-components

In this paper we supply one link in a chain of results which will prove the following two conjectures: B(G)-Conjecture. If H is a 2-local subgroup of a finite group G, then [L(H), O(H)] ⫅ (G). Unbalanced Group Conjecture. If G is a finite group with O(C_G(t)) ⫋ O(G) for some involution t ∈ G, then O(C_G(t)) acts nontrivially on L/Z*(L) where L is a 2-component of G with L/Z*(L) isomorphic to one of the following simple groups: (1) A simple Chevalley group or twisted variation over a field of odd order; (2) An alternating group of odd degree; (3) PSL (3, 4) of He, the simple group of Held.