Corollary

Let $e$ be a framing of $(G,T)$ and let E be the group of automorphisms of G that leave $e$ stable. Then Aut(G) is the semi-direct product of E by Int(G), and Aut(G$,T$) is the semi-direct product of E by Int(G) $\cap$ Aut(G$,T$) $=$ Int$_G(N_G(T))$.

Indeed, every element of Aut(G) transforms $e$ into a framing of G. By Prop. 19, every coset of Aut(G) modulo Int(G) meets E in a single point, hence the first assertion. The second is proved in the same way.