  Let $F$ be a non-Archimedean local field and $G$ be the general linear group
$\mathrm{GL}_n$ over $F$. Bushnell and Henniart described the essentially tame
local Langlands correspondence of $G(F)$ using rectifiers, which are certain
characters defined on tamely ramified elliptic maximal tori of $G(F)$. They
obtained such result by studying the automorphic induction character identity.
We relate this identity to the spectral transfer character identity, based on
the theory of twisted endoscopy of Kottwitz, Langlands, and Shelstad. In this
article, we establish the following two main results. (i) To show that the
automorphic induction character identity is equal to the spectral transfer
character identity, when both are normalized by the same Whittaker data. (ii)
To express the essentially tame local Langlands correspondence using admissible
embeddings constructed by Langlands-Shelstad $\chi$-data, and to relate
Bushnell-Henniart's rectifiers to certain transfer factors.
