Jordan property for automorphism groups of compact spaces in Fujiki's class $C$

Let $X$ be a compact complex space in Fujiki's Class $C$. We show that the group $Aut(X)$ of all biholomorphic automorphisms of $X$ has the Jordan property: there is a (Jordan) constant $J = J(X)$ such that any finite subgroup $G\le Aut(X)$ has an abelian subgroup $H\le G$ with the index $[G:H]\le J$. This extends, with a quite different method, the result of Prokhorov and Shramov for Moishezon threefolds.