  Free and cofree equivariant spectra are important classes of equivariant
spectra which represent equivariant cohomology theories on free equivariant
spaces. Greenlees-Shipley and Pol and the author have given an algebraic model
for rational (co)free equivariant spectra. In this paper, we extend this
framework by proving that the Quillen functors of
induction-restriction-coinduction between categories of (co)free rational
equivariant spectra correspond to Quillen functors between the algebraic models
in the case of connected compact Lie groups. This is achieved using new
abstract techniques regarding correspondences of Quillen functors along Quillen
equivalences, which we expect to be of use in other applications.
