  We extend the construction of multiplicative representations for a free group
G introduced by Kuhn and Steger (Isr. J., (144) 2004) in such a way that the
new class Mult(G) so defined is stable under taking the finite direct sum,
under changes of generators (and hence is Aut(G)-invariant), under restriction
to and induction from a subgroup of finite index.
  The main tool is the detailed study of the properties of the action of a free
group on its Cayley graph with respect to a change of generators, as well as
the relative properties of the action of a subgroup of finite index after the
choice of a "nice" fundamental domain.
  These stability properties of Mult(G) are essential in the construction of a
new class of representations for a virtually free group (Iozzi-Kuhn-Steger,
arXiv:1112.4709v1)
