I posted the following to stackexchange a while ago , without any answers. Maybe the question is too unmotivated, but it seems very natural to me.
Suppose $X$ is a topological space and $G$ a finite group acting on it. We can form the singular complex $C_\bullet(X),$ and then taking homology gives singular homology: $H_*(X) = h_* C_\bullet(X).$ Since $X$ comes with a $G$-action, we could look at the fixed points of this action: $H_*(X^G) = H_* C_\bullet(X^G).$ By functoriality of the singular complex, $C_\bullet(X)$ also affords a $G$-action, and we can take its fixed point homology: $H'_* = h_* (C_\bullet(X)^G).$ Finally, we could also take the fixed points in homology: $H_*(X)^G.$
My question is: how does $H'_*$ (homology of the fixed points of the singular complex) relate to other more typical invariants of $X,$ such as e.g. the ones displayed above ($H_*(X),$ $H_*(X^G),$ $H_*(X)^G$)?