What I am trying to do is to compute $\mathbb{Z}$-graded Bredon homology of $S(1+\sigma)$ over $Q\times\Sigma_2$, where

- $Q$ is a cyclic group of order 2
- $\sigma$ is its real sign representation
- $\Sigma_2$ is acting on $S(1+\sigma)$ by antipodal action.

In particular, different notation for a cyclic group of order 2 is used to emphasize different actions. To make notation more legible, let's put $S:=S(1+\sigma)$. Its Bredon homology with $\underline{\mathbb{F}}_2$ coefficients is given by $H^{Q\times\Sigma_2}_0(S;\underline{\mathbb{F}}_2)=\mathbb{F}_2\oplus\mathbb{F}_2$ and $H^{Q\times\Sigma_2}_1(S;\underline{\mathbb{F}}_2)=\mathbb{F}_2$.

I may compute this homology also by the Universal Coefficient Spectral Sequence, which is given by $$ E^2_{p,q}=Tor^{\mathcal{O}_{Q\times\Sigma_2}}_p\left(H_q(S^\bullet),\underline{\mathbb{F}}_2\right)\Rightarrow H^{Q\times\Sigma_2}_{p+q}(S;\underline{\mathbb{F}}_2) $$

Singular homology here is taken with integral coefficients. (Using this method is a little bit of overkill, but this is supposed to be a toy example for further calculations).

I have inputs of the $E^2$ page computed - they are given by: $$ Tor_i(H_0(S^\bullet),\underline{\mathbb{F}}_2)= \left\{ \begin{array}{cc} \mathbb{F}_2\oplus\mathbb{F}_2 & i=0 \\ \mathbb{F}_2 & i=1 \\ \mathbb{F}_2^{i-1} & i>1 \end{array} \right. $$

and

$$ Tor_i(H_1(S^\bullet),\underline{\mathbb{F}}_2)= \mathbb{F}_2^{i+1} $$ for $i\geq 0$. Everything else vanish.

So I have my second page of the spectral sequence, but I am struggling with identifying the differentials. From the calculations "by hand" I see that they are supposed to be isomorphisms. But I have got no clue how to show that.

Could anybody give me a hand on that?