Let $T$ be the torus over $\mathbb{Q}$ defined by the equation $x^2+y^2=1$, then it becomes isomorphic to $\mathbb{G}_m$ over field $\mathbb{Q}[i]$. Let $X$ denote the character group of $T$, regarded as a locally constant sheaf for the etale topology or the fppf topology over $\mathrm{Spec}\mathbb{Q}$.
Now let's work in the fppf site of $\mathrm{Spec}\mathbb{Q}$, consider the Hom sheaf $\mathscr{H}om(X,\mathbb{G}_a)$, it seems for me that:
this sheaf has trivial global sections;
it becomes isomorphic to $\mathbb{G}_a$ over $\mathrm{Spec}\mathbb{Q[i]}$,
My questions are 1. Is this sheaf represented by a group scheme?
If the answer to 1 is no, is it representable by an algebraic space (something like $\mathbb{G}_a/(\mathbb{Z}/2)$)?
If it is indeed something like $\mathbb{G}_a/(\mathbb{Z}/2)$ ,
how to compute the fppf-cohomology of the algebraic space associated to an action of a finite group on a group scheme? e.g. $H^1_{\mathrm{fppf}}(\mathbb{Q},\mathscr{H}om(X,\mathbb{G}_a))$