$K$-theory of regular compactification bundles

Let $G$ be a connected reductive algebraic group. Let $$\mathcal{E}$\rightarrow $\mathcal{B}$$beaprincipal$G\times G$-bundleand$X$ be a regular compactification of $G$. We describe the Grothendieck ring of the associated fibre bundle $\mathcal{E}(X):=\mathcal{E}{\times }_{G\times G}X$, as an algebra over the Grothendieck ring of a canonical toric bundle over a flag bundle on $\mathcal{B}$. These are relative versions of the results on equivariant $K$-theory of regular compactifications of $G$. They also generalize the well known results on the Grothendieck rings of projective bundles, toric bundles and flag bundles.