Exponent of a finite group with a four-group of automorphisms

Let A be a group isomorphic with either S ₄, the symmetric group on four symbols, or D ₈, the dihedral group of order 8. Let V be a normal four-subgroup of A and ?? an involution in ${A\setminus V}$ . Suppose that A acts on a finite group G in such a manner that C _G (V)?=?1 and C _G (??) has exponent e. We show that if ${A\cong S_4}$ then the exponent of G is e-bounded and if ${A\cong D_8}$ then the exponent of the derived group G?? is e-bounded. This work was motivated by recent results on the exponent of a finite group admitting an action by a Frobenius group of automorphisms.