Proposition 17

The homomorphism Aut(G$,T$)$\rightarrow$ Aut(D) is surjective, with kernel Int$_G(T)$.

Note that Aut(G$,T$)$\cap$ Int(G) = Int$_G(N_G(T))$ and that the image of Int$_G(N_G(T))$ in Aut(D) is W(D) (no. 5, Prop. 10). Thus, Proposition 17 gives an isomorphism

Aut(G$,T$)$/$(Aut(G$,T$)$\cap$ Int(G)) $\rightarrow$ Aut(D)$/W(D)$.

Further, Aut(G) = Int(G).Aut(G$,T$). Indeed, if $u$ belongs to Aut(G), $u(T)$ is a maximal torus of T, hence is conjugate to T, and there exists an inner automorphism $v$ of G such that $u(T) =v$(T), in other words $v^{-1}u\in$ Aut(G$,T$). It follows that Aut(G)$/$Int(G) can be identified with Aut(G$,T$)$/$(Aut(G$,T$)$\cap$Int(G)), so in view of the preceding we have an exact sequence

$1\rightarrow$ Int(G) $\rightarrow$ Aut(G) $\rightarrow$ Aut(D)$/W(D)\rightarrow 1$. (16)

Consequently: