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Using the Dadarlat isomorphism, we give a characterization for the Kasparov product of 
${{C}^{*}}$-algebra extensions. A certain relation between 
$KK\left( A,\,Q\left( B \right) \right)$ and 
$KK\left( A,\,Q\left( KB \right) \right)$ is also considered when 
$B$ is not stable, and it is proved that $KK\left( A,\,Q\left( B \right) \right)$ and $KK\left( A,\,Q\left( KB \right) \right)$ are not isomorphic in general.
