Enhanced group associated to a polarized quaternion order (awaiting review)

Let $(O,\mu)$ be a polarized quaternion order. We define $\widehat{O} \colonequals O\otimes_\Z \widehat{\Z}$. Then $\widehat{O}^\times$ is a profinite group with \[ \widehat{O}^\times = \lim_{\xleftarrow[N]{}} \modstar{O}{NO}, \] where $N$ ranges over all positive integers. We define \[ \Aut_{\pm\mu}(O) \colonequals \{\gamma \in N_{B^\times}(O)/\Q^\times : \gamma^{-1}\mu\gamma = \pm \mu\}. \] Then $\Aut_{\pm\mu}(O)$ acts on $\widehat{O}^\times$. The enhanced group (associated to $(O,\mu)$) is the semidirect product $\Aut_{\pm\mu}(O)\ltimes \widehat{O}^\times$.