  In a previous paper, the authors defined an equivariant version of the
so-called Saito duality between the monodromy zeta functions as a sort of
Fourier transform between the Burnside rings of an abelian group and of its
group of characters. Here a so-called enhanced Burnside ring $\widehat{B}(G)$
of a finite group $G$ is defined. An element of it is represented by a finite
$G$-set with a $G$-equivariant transformation and with characters of the
isotropy subgroups associated to all points. One gives an enhanced version of
the equivariant Saito duality. For a complex analytic $G$-manifold with a
$G$-equivariant transformation of it one has an enhanced equivariant Euler
characteristic with values in a completion of $\widehat{B}(G)$. It is proved
that the (reduced) enhanced equivariant Euler characteristics of the Milnor
fibres of Berglund-H\"ubsch dual invertible polynomials coincide up to sign and
show that this implies the result about orbifold zeta functions of
Berglund-H\"ubsch-Henningson dual pairs obtained earlier.
