Let us consider the fibration
$$ M\hookrightarrow EG\times_{\varphi}M\twoheadrightarrow BG $$ where $M$ is a $G-$space, $\varphi$ is the action of $G$ on $M$ and $EG\times_{\varphi}M$ is the diagonal quotient. It is required that $M$ is compact and path connected, $G$ is discrete and the action fixes at least one point $pt$. I would like to compute the equivariant cohomology $H_{G}^{\bullet}(M,\mathbb{Z})$. To do that, I can get the following information. Let us consider the Leray-Serre spectral sequence associated to the given fibration $$ E_{2}^{p,q}=H^{p}(BG,H^{q}(M,\mathbb{Z})) $$ and let us consider the terms $E_{2}^{p,0}$. In this case, because $M$ is path connected $H^{0}(M,\mathbb{Z})\simeq\mathbb{Z}$ and hence $$ E_{2}^{p,0}=H^{p}(BG,\mathbb{Z})\simeq H_{G}^{p}(pt,\mathbb{Z}) $$ because the action fixes the point $pt$. Introducing the reduced cohomology group $$ \widetilde{H}_{G}^{p}(M,\mathbb{Z}):=\ker i^{p} $$ where $i^{p}:H_{G}^{p}(M,\mathbb{Z})\twoheadrightarrow H_{G}^{p}(pt,\mathbb{Z})$ is the map in cohomology induced by the inclusion $i:pt\hookrightarrow M$, we have that $$ H_{G}^{p}(M,\mathbb{Z})\simeq\widetilde{H}_{G}^{p}(M,\mathbb{Z})\oplus H_{G}^{p}(pt,\mathbb{Z})\simeq\widetilde{H}_{G}^{p}(M,\mathbb{Z})\oplus E_{2}^{p,0}. $$ Thus $E_{2}^{p,0}$ is a subgroup of $H_{G}^{p}(M,\mathbb{Z})$. From that, I would like to deduce that $$ E_{2}^{p,0}=E_{\infty}^{p,0}\qquad\forall p\geqslant0, $$ i.e., $\textrm{Im}\,d_{2}^{p-2,1}=0$ for every $p\geqslant0$. I can not see why I can affirm that. Do you have some suggestions? Thanks in advance.
I think that the following argument works:
Firstly, I am gonna recall general property of spectral sequence for fiber bundle
I am gonna use letter T for total space of fiber bundle and letter B for base of this bundle (I also asume that this fuber bundle is cohomologically simple)
Map $H^{n}(B;\mathbb{Z}) = E^{n,0}_{2} \rightarrow E^{n,0}_{3} \rightarrow \dots \rightarrow E^{n,0}_{\infty} \subset H^{n}(T; \mathbb{Z})$
coincides with map induced by projection $\pi: T \rightarrow B$
This can be deduced from naturality of Serre spectral sequence applied to map from fiber bundle $T\xrightarrow{\mathrm{id}} T$ to $T\xrightarrow{\pi} P$.
In particular, if homomorphism $\pi^{*}$ is injective then $E^{n,0}_{2} = E^{n,0}_{\infty}$.
Now, back to your specific spectral sequences associated to fiber bundle $EG\times_{\varphi} M \xrightarrow{\pi} EG\times_{\varphi}pt = BG$. We need to show that $\pi^{*}$ is injective. To prove that consider equivariant map $i:pt \rightarrow M$ which is embedding of fixed point in manifold $M$. As you mentioned, this map induce map $i_{G}: EG\times_{\varphi}pt \rightarrow EG\times_{\varphi}M$.
Note that composition $\pi\circ i_{G} = \mathrm{id}$, so we have equality $i_{G}^{*}\circ \pi^{*} = \mathrm{id}$. From this equality one can conclude that $\pi^{*}$ is injective and we are done.