Fixed point indices of equivariant maps of certain Jiang spaces

Given $X$ a Jiang space we know that all Nielsen classes have the same index. Now let us consider $X$ a $G$-space where $G$ is a finite group which acts freely on $X$. In [P. Wong, < i> Equivariant Nielsen numbers< /i> , Pacific J. Math. < b> l59< /b> (1993), 153–175], we do have the notion of $X$ to be an equivariant Jiang space and under this condition it is true that all equivariant Nielsen classes have the same index. We study the question if the weaker condition of $X$ being just a Jiang space is sufficient for all equivariant Nielsen classes to have the same index. We show a family of spaces where all equivariant Nielsen classes have the same index. In many cases the spaces of such a family are not equivariant Jiang spaces. Finally, we also show an example of one Jiang space together with equivariant maps where the equivariant Nielsen classes have different indices.