  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 $G'$ 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.
