Suppose I have a trivial vector bundle $V\cong \mathcal{O}_C^{\oplus s} \rightarrow C$ on an algebraic variety $C$, and suppose furthermore that I have an action $\mu$ of a cyclic finte group $G$ on the vector bundle $V$ that gives a decomposition of $V$ into eigenbundles, indexed by the characters of $G$. Are the eigenbundles trivial as well? Why?

No. Let $C$ be an open affine part of an elliptic curve over the complex numbers and $L$ a nontrivial line bundle on $C$. Now, $L\oplus L^{1}$ is trivial, so let $G=\mathbf Z/2\mathbf Z$ operate by $1$ on $L$ and by $1$ on $L^{1}$. For complete varieties (for simplicity, say having a rational point), all endomorphisms of a free bundle are given by constant matrices, so all direct summands are free again. 

