  Given two automorphisms of a group $G$, one is interested in knowing whether
they are conjugate in the automorphism group of $G$, or in the abstract
commensurator of $G$, and how these two properties may differ. When $G$ is the
fundamental group of a closed orientable surface, we present a uniform
finiteness theorem for the class of pseudo-Anosov automorphisms. We present an
explicit example of a commensurably conjugate pair of pseudo-Anosov
automorphisms of a genus $3$ surface, that are not conjugate in the Mapping
Class Group, and we also show that infinitely many independent automorphisms of
hyperbolic orbifolds have class number equal to one. In the appendix, we
briefly survey the Latimer-MacDuffee theorem that addresses the case of
automorphisms of $\mathbb{Z}^n$, with a point of view that is suited to an
analogy with surface group automorphisms.
