Torsors trivializing over a fixed finite etale cover

Question

Let $S$ be an integral regular scheme and let $T\to S$ be a finite etale morphism. Let $G$ be a smooth affine finite type group scheme over $S$.

Is the set of $S$-isomorphism classes of $G$-torsors over $S$ which are trivial over $T$ finite?

I guess the set-up is ridiculously general. So what if

1) $S$ is of finite type over $\mathbb C$, or

2) $S$ is of finite type $\mathbb F_p$, or

4) $S$ is of finite type over $\mathbb Z_p$, or

5) $S$ is of finite type over $\mathbb Q_p$, or

6) $S$ is of finite type over $\mathbb Z$?

Answer 1

The answer to the general question is NO.

We take $S=\mathrm{Spec}\, K$, where $K$ is a number field. Let $L/K$ be a finite Galois extension. We set $T=\mathrm{Spec}\, L$. We consider the norm homomorphism $$N_{L/K}\colon R_{L/K}\mathbb{G}_{m,L}\to \mathbb{G}_{m,K}\,,$$ where $R_{L/K}$ denotes the Weil restriction of scalars. Set $G=\mathrm{ker}\, N_{L/K}$. Then our torus $G$ splits over $L$, hence $H^1(L,G)=1$ by Theorem 90. From the short exact sequence $$1\to G\to R_{L/K}\mathbb{G}_{m,L}\to \mathbb{G}_{m,K}\to 1$$ and the induced Galois cohomology exact sequence $$ L^*\to K^* \to H^1(K,G)\to H^1(K,R_{L/K}\mathbb{G}_{m,L})=1$$ we obtain that $H^1(K,G)= K^*/ N_{L/K}(L^*)$, and this group is infinite. Thus the kernel $$\mathrm{ker}[H^1(K,G)\to H^1(L,G)]=H^1(K,G)= K^*/ N_{L/K}(L^*)$$ is infinite, as required.