My question is as follows:

Let $k$ be a field of characteristic zero and let $\overline{k}$ be an algebraic closure. Let $V$ be an algebraic variety over $k$ and let $\overline{V}=V \times_k \overline{k}$. Suppose that $\overline{V}$ admits the structure of an algebraic group. Then is $V$ itself a principal homogeneous space for

somealgebraic group?

Note that the answer to my question is *yes* when $V$ is projective; in this case it is well-known that such a $V$ is a principal homogeneous space for its Albanese variety.
So I am really interested in the case where $V$ is affine. Here I am not aware of an analogue of the Albanese variety for linear algebraic groups.

Even if the answer is *no* in general, I would still be interested in some positive results for special cases, e.g. for reductive groups or semisimple groups.