Equivalence of ${\mathbb{Z}}_4$-actions on handlebodies of genus $g$

In this paper we consider all orientation-preserving ${\mathbb{Z}}_4$-actions on $3$-dimensional handlebodies $V_g$ of genus $g>0$. We study the graph of groups $(\mathrm{\Gamma }($v$),\mathbf{G}(\mathbf{v}))$, which determines a handlebody orbifold $V(\mathrm{\Gamma }($v$),\mathbf{G}(\mathbf{v}))\simeq V_g\mathrm{/}{\mathbb{Z}}_4$. This algebraic characterization is used to enumerate the total number of ${\mathbb{Z}}_4$ group actions on such handlebodies, up to equivalence.