On the exponent of a finite group with an automorphism group of order twelve

Suppose that a Frobenius group FH with kernel F of order 4 and complement H of order 3 acts coprimely on a finite group G in such a manner that CG(F)=1 and CG(H) has exponent e. We prove that the exponent of G is e-bounded. Earlier results of this nature were known only for metacyclic Frobenius groups.