Two different local Langlands parameters for quadratic extension

Question

Let $E/F$ be a quadratic extension of nonarchimedean local field, and let $G$ be a reductive group over $E$, and $G(E)$ the $E$-points of $G$. (For my question, one may just assume $G=\operatorname{GL}_n$.) Now $G(E)$ can be viewed in two different way: the $F$ points of the restriction of scalar $G_1:=\operatorname{Res}_{E/F}G$, and the $E$-points of just $G_2:=G$. Their Langlands $L$-groups are respectively $$ {^LG_1}=(G^\vee(\mathbb{C})\times G^\vee(\mathbb{C}))\rtimes W_F\qquad\text{and}\qquad{^LG_2}=G^\vee(\mathbb{C})\times W_E, $$ where $G^\vee$ is the dual group of $G$ and for the semidirect product the nontrivial element in $W_F/ W_E$ acts by switching the two factors.

Now, assuming the local Langlands correspondence, for each irreducible admissible representation of $G(E)$, one can associate local Langlands parameters $$ \varphi_1:\operatorname{WD}_F\longrightarrow{^LG_1}\qquad\text{and}\qquad \varphi_2:\operatorname{WD}_E\longrightarrow{^LG_2}, $$ where $\operatorname{WD}_F$ and $\operatorname{WD}_E$ are the Weil-Deligne groups.

I believe there must be a way to pass from $\varphi_1$ to $\varphi_2$. But how?

Answer 1

This came up in a paper of mine not so long ago, and my coauthors and I were surprised that it wasn't made explicit in the standard references, so we wrote it out ourselves:

Dembélé, Lassina; Loeffler, David; Pacetti, Ariel, Non-paritious Hilbert modular forms, Math. Z. 292, No. 1-2, 361-385 (2019). ZBL1446.11084.

See Proposition 1.3. (Strictly speaking we are describing the analogue for global fields, not local fields, but the recipe is the same.)