We know that $\left\vert G\right\vert <\infty \Rightarrow \left\vert Aut\left( G\right) \right\vert <\infty $.
My question is: $\exp \left( G\right) <\infty \Rightarrow \exp \left( Aut\left( G\right) \right) <\infty ?$
Where $Aut\left( G\right) $=set automorphisms of $G$ and $\exp \left( G\right)$ is the exponent of G.