  First we prove that any inner automorphism in the stabilizer of a
graded-simple unital associative algebra whose grading group is abelian is the
conjugation by a homogeneous element. Now consider a grading by an abelian
group on an associative algebra such that the algebra is graded-simple and
satisfies the DCC on graded left ideals. We give necessary and sufficient
conditions for the grading to be fine. Then we assume that one of these
necessary conditions to be fine is satisfied, and we compute the automorphism
groups of the grading; the results are expressed in terms of the automorphism
groups of a graded-division algebra. Finally we compute the automorphism groups
of graded-division algebras in the case in which the ground field is the field
of real numbers, and the underlying algebra (disregarding the grading) is
simple and of finite dimension.
