  In this paper, the extended Musielak-Orlicz-Gauss image problem is studied.
Such a problem aims to characterize the Musielak-Orlicz-Gauss image measure
$\widetilde{C}_{G,\Psi,\lambda}(\Omega,\cdot)$ of convex body $\Omega$ in
$\mathbb{R}^{n+1}$ containing the origin (but the origin is not necessary in
its interior). In particular, we provide solutions to the extended
Musielak-Orlicz-Gauss image problem based on the study of suitably designed
parabolic flows, and by the use of approximation technique (for general
measures). Our parabolic flows involve two Musielak-Orlicz functions and hence
contain many well-studied curvature flows related to Minkowski type problems as
special cases. Our results not only generalize many previously known solutions
to the Minkowski type and Gauss image problems, but also provide solutions to
those problems in many unsolved cases.
