  In this paper, we study the relations between groups related to cluster
automorphism groups which are defined by Assem, Schiffler and Shamchenko in
\cite{ASS}. We establish the relationship among (strict) direct cluster
automorphism groups and those groups consisting of periodicities of
respectively labeled seeds and exchange matrices in the language of short exact
sequences. As an application, we characterize automorphism-finite cluster
algebras in the cases with bipartite seeds or finite mutation type. Finally, we
study the relation between the groups $\mathrm{Aut}\mathcal{A}$ and
$\mathrm{Aut}_{M_n}S$ and give the negative answer via counter-examples to King
and Pressland's a problem in \cite{KP}.
