  We construct a Hopf action, with an invariant trace, of a bicrossed product
Hopf algebra $\cH=\big( \cU(\Fg_1) \acr \cR(G_2) \big)^{\cop}$ constructed from
a matched pair of Lie groups $G_1$ and $G_2$, on a convolution algebra
$\cA=C_c^{\ify}(G_1)\rtimes G_2^{\delta}$. We give an explicit way to construct
Hopf cyclic cohomology classes of our Hopf algebra $\cH$ and then realize these
classes in terms of explicit representative cocycles in the cyclic cohomology
of the convolution algebra $\cA$.
