Let $X$ be a $G$-space. Are there examples, i.e. conditions or classes of spaces, such that the map induced by the inclusion $X^G\to X$ of the fixed part into the whole space induce a surjection of equivariant homologies $H_{\ast}^G(X^G)\to H_{\ast}^G(X)$ (in all dimensions)?

Now let $X$ be a pointed $G$-space. Define the half smash $EG\ltimes_G X$ as $(EG\times_G X)/(EG\times_G \ast)$. Again, are there examples where the map $H_{\ast}(EG\ltimes_G X^G)\to H_{\ast}(EG\ltimes_G X)$ is a surjection?

Finally, let $Ci(X)$ be the cofibre of the inclusion $i:X\to EG\ltimes_G X$. When is $H_{\ast}(Ci(X^G))\to H_{\ast}(Ci(X))$?