For any finite group G, we impose an algebraic condition, the Gnil-coset condition, and prove that any finite Oliver group G satisfying the Gnil-coset condition has a smooth action on some sphere with isolated fixed points at which the tangent G-modules are not isomorphic to each other. Moreover, we prove that, for any finite non-solvable group G not isomorphic to Aut(A6) or PΣL(2, 27), the Gnil-coset condition holds if and only if rG ≥ 2, where rG is the number of real conjugacy classes of elements of G not of prime power order. As a conclusion, the Laitinen Conjecture holds for any finite non-solvable group not isomorphic to Aut(A6).
