Here are a few recent results which haven't been mentioned so far (in chronological order).

- Rajan, On the image and fibres of solvable base change (2002)

This relies on the result of Lapid and Rogawski for $\mathrm{GL}(2)$. The paper works with the generalization for $\mathrm{GL}(n)$ as a hypothesis. Remark 1 there says that, "Granting this, our theorem extends to $\mathrm{GL}_n$, and we present the proof in the general case assuming Statement B of Lapid-Rogawski."

- Henniart, Induction automorphe pour $\mathrm{GL}(n,\mathbb{C})$ (2009)

The result is the same as that obtained in the non-archimedean case.

- Henniart, Lemaire, Formules de caracteres pour l'induction automorphe (2010)

If $E/F$ is a degree $d$ cyclic extension of non-archimedean local fields then Henniart and Herb proved that if $\tau$ is a tempered representation of $\mathrm{GL}(m,E)$ then there is a tempered representation $\pi$ of $\mathrm{GL}(md,F)$ such that
\begin{equation}
\mathrm{tr}\,\tau(f)=c\cdot\mathrm{tr}\,\pi(f^G)\circ A,
\end{equation}
whenever $f$ and $f^G$ have matching orbital integrals. The constant $c$ depends on $\tau$ and the operator $A$ which intertwines $\pi$ and $\omega\pi$, where $\omega$ is a character of $F^\times$ determining the extension $E/F$. Henniart and Lemaire show that when one uses the canonical normalization of $A$ via Whittaker models then $c$ does not depend on $\tau$.

- Hiraga and Ichino, On the Kottwitz-Shelstad normalization of transfer factors for automorphic induction for $\mathrm{GL}_n$ (2012)

They show that this constant $c$ is in fact 1 when using normalized transfer factors.

- Henniart, Induction automorphe globale pour les corps de nombres (2012)

When $E/F$ is a cyclic extension of number fields of degree $d$ and $\tau$ is an induced from cuspidal representation of $\mathrm{GL}(m,\mathbb{A}_E)$, from the abstract,

"We prove that the representation $\pi$ automorphically induced from $\tau$ exists, and we study the fibres and the image of automorphic induction. For that we use and extend the results of Arthur and Clozel on base change, which corresponds to restricting Galois representations from $F$ to $E$, and we clarify the relations between the two processes. Moreover we prove that global automorphic induction is compatible, at finite places, with the local automorphic induction defined by R. Herb and the author."