# Two different local Langlands parameters for quadratic extension

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?