For [Formula: see text] a finite group and [Formula: see text] a [Formula: see text]-space on which a normal subgroup [Formula: see text] acts trivially, we show that the [Formula: see text]-equivariant [Formula: see text]-theory of [Formula: see text] decomposes as a direct sum of twisted equivariant [Formula: see text]-theories of [Formula: see text] parametrized by the orbits of the conjugation action of [Formula: see text] on the irreducible representations of [Formula: see text]. The twists are group 2-cocycles which encode the obstruction of lifting an irreducible representation of [Formula: see text] to the subgroup of [Formula: see text] which fixes the isomorphism class of the irreducible representation.