Reduced Equivariant Cohomology

Question

I read an article stating that if a finite group $G$ acts on a compact and path connected topological space $M$ and the action has a fixed point $x_0$, then $$ H^p_G(M;\mathbb{Z})\simeq H^p_G(x_0;\mathbb{Z})\oplus \widetilde{H}^p_G(M;\mathbb{Z})\qquad\forall p\qquad\qquad(1) $$ where $H^p_G(M;\mathbb{Z})$ is the Equivariant Cohomology group of $M$ and it is written that $\widetilde{H}^p_G(M;\mathbb{Z})$ is the reduced cohomology group. Since $$ H^p_G(x_0;\mathbb{Z})\simeq H_{\text{group}}^p(G;\mathbb{Z}), $$ and for certain groups, such as cyclic and dihedral groups, $H_{\text{group}}^p(G;\mathbb{Z})$ can be different from zero also for $p>0$, it seems that the definition of reduced cohomology in equivariant cohomology may differ from the definition given in classical cohomology (where $H^p(M;\mathbb{Z})=\widetilde{H}^p(M;\mathbb{Z})$ for $p>0$). Based on equation (1), it seems that $\widetilde{H}^p_G(M;\mathbb{Z})$ is the cokernel of the map $$ H^p_G(x_0;\mathbb{Z})\rightarrow H^p_G(M;\mathbb{Z}) $$ induced by the constant map $M\rightarrow x_0$. Is this understanding correct? If so, why is there a different definition of reduced cohomology in equivariant cohomology? Can you provide any references on this topic? Thank you in advance.