  A linear code is called an MDS self-dual code if it is both an MDS code and a
self-dual code with respect to the Euclidean inner product. The parameters of
such codes are completely determined by the code length. In this paper, we
consider new constructions of MDS self-dual codes via generalized Reed-Solomon
(GRS) codes and their extended codes. The critical idea of our constructions is
to choose suitable evaluation points such that the corresponding (extended) GRS
codes are self-dual. The evaluation set of our constructions is consists of a
subgroup of finite fields and its cosets in a bigger subgroup. Four new
families of MDS self-dual codes are obtained and they have better parameters
than previous results in certain region. Moreover, by the Mobius action over
finite fields, we give a systematic way to construct self-dual GRS codes with
different evaluation points provided any known self-dual GRS codes. Specially,
we prove that all the self-dual extended GRS codes over $\mathbb{F}_{q}$ with
length $n< q+1$ can be constructed from GRS codes with the same parameters.
