descent in derived category

Question

Suppose we are given a field $k$ and a finite Galois extension $L$ with Galois group $G$. We consider the projection $\pi:X\otimes_k L\rightarrow X$ for a smooth projective variety $X$. The object $\pi^*\pi_*\mathcal{F}$ for a coherent sheaf $\mathcal{F}$ should be $\bigoplus_{\lambda}\mathcal{F}^{\lambda}$ where $\mathcal{F}^{\lambda}$ is the Galois conjugate of $\mathcal{F}$ and $\lambda\in G$. I now have two questions:

1. Suppose $\mathrm{Ext}^i(\mathcal{F},\mathcal{F})=0$ for $i>0$. Does this imply $\mathrm{Ext}^i(\mathcal{F},\mathcal{F}^{\lambda})=0$ for $i>0$?

2. What happens in the derived world i.e. for a given object $\mathcal{K}$ in $D^b(X\otimes_k L)$ what can one say about $L\pi^*R\pi_*\mathcal{K}$ ?

Answer 1

For 2): the same decomposition holds in the derived category as well: note that $\pi $ is flat and affine, so $L\pi ^*$ and $R\pi _*$ of a complex are just obtained by applying $\pi ^*$ and $\pi _*$ to each component.

I don't see any reason for 1) to be true but it is not so easy to think of a counter-example...