  We develop a fundamental theory of compact quantum group equivariant finite
extensions of C*-algebras. In particular we focus on the case of quantum
homogeneous spaces and give a Tannaka-Krein type result for equivariant
correspondences. As its application, we show that every Jones' value appears as
the index of an equivariant conditional expectation. In the latter half of this
paper, we give an imprimitivity theorem in some cases: for general compact
quantum groups under a finiteness conditions, and for the Drinfeld-Jimbo
deformation $G_q$ of a simply-connected compact Lie group $G$. As an
application, we give a complete classification of finite index discrete quantum
subgroups of $\widehat{G_q}$.
