$\DeclareMathOperator{\Ind}{Ind}\DeclareMathOperator{\Gal}{Gal}$Let $F$ be a $p$-adic field, and let $E \subseteq \overline{F}$ be a finite extension of $F$. Let $k_F$ and $k_E$ be the residue fields of $F$ and $E$, so $k_F \subseteq k_E$. Let $I_F \subseteq W_F \subseteq \Gal(\overline{F}/F)$ be the inertia and Weil groups of $F$, and define the Weil group of the finite field $k_F$ by $W_{k_F} := W_F/I_F \subseteq \Gal(\overline{k_F}/k_F)$.
If $(\rho,V)$ is a finite dimensional representation of $W_E$, then $$\rho^{I_E}=V^{I_E} = \{v \in V : \rho(x)v = v \textrm{ for all } x \in I_E\}$$ is well defined as a representation of $W_E/I_E = W_{k_E}$.
Consider the two representations
$$\big(\Ind_{W_E}^{W_F}\rho \big)^{I_F} \space \space \space \space \space \Ind_{W_{k_E}}^{W_{k_F}}\rho^{I_E}$$
of $W_{k_F}$. Consider an element $f$ of the left hand side. It is a function $f: W_F \rightarrow V$ satisfying
$$f(yx) = \rho(y)f(x) \tag{$y \in W_E, x \in W_F$}$$
$$f(xz) = f(x) \tag{$x \in W_F, z \in I_F$}$$
Assume that $E/F$ is Galois, then $I_F \cap W_E = I_E$ is normal in $W_F$, which implies that
$$\rho(y)f(x) = f(yx) = f(x(x^{-1}yx)) = f(x)$$
for all $y \in I_E, x \in W_F$. Hence $f$ actually maps into $V^{I_E}$, which gives us a map from the left hand side to the right hand side.
Deligne (Prop. 3.8.1, Les Constants des Equations Fonctionelles des Fonctiones L) indicates that these two representations of $W_{k_F}$ are always isomorphic. But I seem to be only able to get a map from one to the other when $E/F$ is Galois. Are these representations really the same?