  We study the structure of the Mordell--Weil groups of semiabelian varieties
over large algebraic extensions of a finitely generated field of characteristic
zero. We consider two types of algebraic extensions in this paper; one is of
extensions obtained by adjoining the coordinates of certain points of various
semiabelian varieties; the other is of extensions obtained as the fixed
subfield in an algebraically closed field by a finite number of automorphisms.
Some of such fields turn out to be new examples of Kummer-faithful fields which
are not sub-$p$-adic. Among them, we find both examples of Kummer-faithful
fields over which the Mordell--Weil group modulo torsion can be free of
infinite rank and not free.
