  The cross coproduct braided group $Aut(C) \rcocross B$ is obtained by
Tannaka-Krein reconstruction from $C^B\to C$ for a braided group $B$ in braided
category $C$. We apply this construction to obtain partial solutions to two
problems in braided group theory, namely the tensor problem and the braided
double. We obtain $Aut(C)\rcocross Aut(C)\isom Aut(C)\lcross Aut(C)$ and higher
braided group `spin chains'. The example of the braided group $B(R)\lcross
B(R)\lcross...\lcross B(R)$ is described explicitly by R-matrix relations. We
also obtain $Aut(C)\rcocross Aut(C)^*$ as a dual quasitriangular `codouble'
braided group by reconstruction from the dual category $C^\circ\to C$. General
braided double crossproducts $B\dcross C$ are also considered.
