Galois cover an affine scheme Let $X = \operatorname{Spec}(A)$ be an affine scheme, with $A$ noetherian (and normal if this is useful). We suppose that $X$ is a finite étale covering of $Y = \operatorname{Spec}(B)$, Galois with group $G$. So the morphism $X \to Y$ comes from $B \cong A^G \hookrightarrow A$.
I know that $A$ is a projective $B$-module, hence locally free. Is it the case that $A$ is always a free $B$-module?

I don't know if this is the simplest example, but it's the one that I thought to try first.
Let $K=\mathbb{Q}(\sqrt{-15})$. Then, $L=\mathbb{Q}(\sqrt{-3},\sqrt{5})$ is the Hilbert class field for $K$ and so, in particular, $\mathcal{O}_L/\mathcal{O}_K$ is unramified. But, $\mathcal{O}_K$ is a Dedekind domain, and thus $\mathcal{O}_L$ is flat over $\mathcal{O}_K$ since it's torsion free. Thus, $\text{Spec}(\mathcal{O}_L)\to\text{Spec}(\mathcal{O}_K)$ is a finite etale cover, where the top scheme is normal and Noetherian. It's Galois since $\text{Aut}(\mathcal{O}_L/\mathcal{O}_K)=\text{Aut}(L/K)$ and $L/K$ is Galois.
That said, $\mathcal{O}_L/\mathcal{O}_K$ is not free. I leave this for you to check. 
I would like to comment that finding examples isn't quite as easy as Martin suggests. All the most basic examples you'd want to write down have $A^G$ in a class of rings where all finite projective modules are free (e.g. local rings, PIDs, polynomial rings over fields, etc.).
