Let C be a locally compact group and Γ be a closed subgroup of the topological product group G × C. In this paper we study the stable characters of Γ- amenable groups

which concern the responsive properties of the homomorphic images

Subgroups and quotient groups of G when G is Γ- amenable. Some relative results of amenable groups and inner amenable groups are extended. In addition

the relation between that G is Γ-amenable and that Γ has the fixed point property is also analysed in this paper.