  We show that for any compact Lie group $G$ with identity component $N$ and
component group $W=G/N$, the category of free rational $G$-spectra is
equivalent to the category of torsion modules over the twisted group ring
$H^*(BN)[W]$. This gives an algebraic classification of rational
$G$-equivariant cohomology theories on free $G$-spaces and a practical method
for calculating the groups of natural transformations between them.
  This uses the methods of arXiv:1101.2511, and some readers may find the
simpler context of the present paper highlights the main thread of the
argument.
