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Self-normalizing Sylow subgroups

Using the classification of finite simple groups we prove the following statement: Let p > 3 be a prime, Q a group of automorphisms of p-power order of a finite group G, and P a Q-invariant Sylow p-subgroup of G. If C NG(P)/P (Q) is trivial, then G is solvable. An equivalent formulation is that if G has a self-normalizing Sylow p-subgroup with p > 3 a prime, then G is solvable. We also investigate the possibilities when p = 3.