Let $K$ be a finite extension of $\mathbb{Q}_p$ and $L$ be a finite Galois extension of $K$. Let $V$ be a $p$adic representation of $G_L$ Let $D$ be the $(\varphi, \Gamma)$module associated to $V$ by Fontaine's functor. Is there an easy way to describe the $(\varphi, \Gamma)$module associated to $Ind_{G_L}^{G_K} V$ in terms of $D$ ?
If $L/K$ is unramified, then the answer is exactly what Matt said. In the general case, you have to take into account the fact that $\Gamma_K$ is larger than $\Gamma_L$ and the construction is given in 2.2 of Ruochuan Liu's "Cohomology and Duality for (phi,Gamma)modules over the Robba ring", see http://arxiv.org/abs/0711.4346 


I've always wondered this too. I think that if $L$ over $K$ is unramified, then we just take the $(\varphi,\Gamma)$module for $V$, which is a $(\varphi,\Gamma)$module over $\mathcal E_L$, and regard it as a $(\varphi,\Gamma)$module over $\mathcal E_K$ (which is a subfield of $\mathcal E_L$). (Probably you already knew this, assuming it is correct.) In the ramified case, I'm not sure; hopefully Laurent Berger will see this and answer! 

