  We construct the crossed product of a C(X)-algebra by an endomorphism, in
such a way that the endomorphism itself becomes induced by the bimodule of
continuous sections of a vector bundle. Some motivating examples for such a
construction are given. Furthermore, we study the C*-algebra of G-invariant
elements of the Cuntz-Pimsner algebra associated with a G-vector bundle, where
G is a (noncompact, in general) group. In particular, the C*-algebra of
invariant elements w.r.t. the action of the group of special unitaries of the
given vector bundle is a crossed product in the above sense. We also study the
analogous construction on certain Hilbert bimodules, called 'noncommutative
pullbacks'.
