Semisimple types for quaternionic forms of p-adic classical groups and compatible beta-extensions

Let $G$ be a quaternionic form of a $p$-adic classical group ($p$ odd).

We construct a Bushnell-Kutzko-Stevens type for every Bernstein block of the category of smooth complex representations of $G$.

Further we construct a system of compatible $\beta$-extensions, i.e. a family of $\beta$-extensions parametrised by the points of a chamber of the Bruhat-Tits building of the centralizer $G_\beta$ which are related via transfer.