  Let $A \subseteq E$ be an extension of Hopf algebras such that there exists a
normal left $A$-module coalgebra map $\pi : E \to A$ that splits the inclusion.
We shall describe the set of all coquasitriangular structures on the Hopf
algebra $E$ in terms of the datum $(A, E, \pi)$ as follows: first, any such
extension $E$ is isomorphic to a unified product $A \ltimes H$, for some
unitary subcoalgebra $H$ of $E$ (\cite{am2}). Then, as a main theorem, we
establish a bijective correspondence between the set of all coquasitriangular
structures on an arbitrary unified product $A \ltimes H$ and a certain set of
datum $(p, \tau, u, v)$ related to the components of the unified product. As
the main application, we derive necessary and sufficient conditions for Majid's
infinite dimensional quantum double $D_{\lambda}(A, H) = A \bowtie_{\tau} H$ to
be a coquasitriangular Hopf algebra. Several examples are worked out in detail.
