Homogeneous spaces of Hilbert type

Let k be a global field. Let G be a connected linear algebraic k-group, assumed reductive when k is a function field. It follows from a result of a preprint by Bary-Soroker, Fehm and Petersen that when H is a smooth connected k-subgroup of G, the quotient space G/H is of Hilbert type. We prove a similar result for certain non-connected k-subgroups H of G. In particular, we prove that if G is a simply connected k-group over a number field k, and H is an abelian k-subgroup of G, not necessarily connected, then G/H is of Hilbert type.