Graded Artin-Wedderburn Theorem and Mackey's obstruction map

Simply-graded F-algebras which are (left) graded-Artinian are well understood. They are described analogously to the classical Artin-Wedderburn Theorem. When F is algebraically closed and the graded simple algebra, or rather its base algebra, is finite-dimensional over it, then a cohomology class appears. We interpret this class as a graded Mackey's obstruction to extending invariant modules from an algebra A to a graded algebra whose base algebra is A.