  We give a Dehn-Nielsen type theorem for the homology cobordism group of
homology cylinders by considering its action on the acyclic closure, which was
defined by Levine, of a free group. Then we construct an additive invariant of
those homology cylinders which act on the acyclic closure trivially. We also
describe some tools to study the automorphism group of the acyclic closure of a
free group generalizing those for the automorphism group of a free group or the
homology cobordism group of homology cylinders.
