Conformal (respectively, anticonformal) automorphisms of the Riemann sphere are provided by the Möbius (respectively, extended Möbius) transformations. A Kleinian group (respectively, an extended Kleinian group) is a discrete group of Möbius transformations (respectively, a discrete group of Möbius and extended Möbius transformations, necessarily containing extended ones). A function group (respectively, an extended function group) is a finitely generated Kleinian group (respectively, a finitely generated extended Kleinian group) with an invariant connected component of its region of discontinuity. A structural decomposition of function groups, in terms of the Klein- Maskit combination theorems, was provided by Maskit in the middle of the 70’s. One should expect a similar decomposition structure for extended function groups, but it seems not to be stated in the existing literature. The aim of this paper is to state and provide a proof of such a decomposition structural picture.