Tensor operators and twisted group algebras

Two generalizations of Frame's theory of the conjugating representation of a finite group G are explored and applied to the problem of forming tensor operators out of a group algebra. In each case the group G acts on the group algebra A (Ḡ), where Ḡ either contains G or covers G as a central extension. By construction, an affirmative answer is given to the question, raised by de Vries and van Zanten, as to whether or not Ḡ may be found such that A (Ḡ) carries all the irreducible representations of G. Several examples are given, with the groups chosen from among the crystallographic point groups.