On the exponent of a finite group admitting a fixed-point-free 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 $\alpha$ 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 $C_G(\alpha )$ 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 $G^{\mathrm{\prime }}$ 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.