Which groups can have $GSp(4)$ as local component? In some cases the relations between a global group $G$ (over the adeles $\mathbb{A}$ of a field $F$) and its local components $G_v$ (where $v$ are the places of $F$) are well known. Obviously a group determines its local components, so the question aims at, given a family of local groups , understanding whether or not it can arises as the local components of global group. Some examples:


*

*the local components of a quaternion algebra are $GL_2(F_v)$ almost everywhere, and an even number of local division quaternion algebras $B_v$

*the local components of a unitary group are $GL_2(F_v)$ at the split places, and any collection of local unitary groups at the other places, maybe safe one (Hasse invariants condition, empty if the number of variables is odd)


Now here is the question:

what natural global group can have $GSp(4)$ as (one, many or every) local component(s)? 

 A: In addition to the relatively boring extension/restriction of scalars for $GSp(2n,\widetilde{k})$ for an extension field $\widetilde{k}$ of the ground field $k$... :
The Galois twist groups often denoted by $Sp^*(p,q)$ (or $GSp^*(p,q)$...) over ground field $\mathbb R$, defined via non-degenerate quaternion-symmetric forms with quaternion algebras over the ground field (or over extensions...) almost everywhere locally become $Sp(p+q,k_v)$, because the quaternion algebra splits almost everywhere locally. (Throwing in the similitudes is minor...)
That is, with quaternion (division) algebra $B$ over ground field $k$ (a global field), with non-degenerate $B$-valued, symmetric form on a $B$-vectorspace $V$ of $B$-dimension $n$, the isometry group $G$ has $k_v$-points isomorphic to $Sp(2n,k_v)$ locally almost everywhere.
EDIT: it may be worth adding that this sort of Galois twisting allows easy creation of compact quotients analogous to Shimura curves for $SL(2)=Sp(2)$, by choosing a symmetric form whose signature at one real place is $(0,q)$ or $(p,0)$, so that (by a standard, if not widely understood, reduction theory result) the arithmetic quotient is compact. But/and the representation theory is just that of (locally, split) $Sp(n)$ almost everywhere.
