I have the following problem. Let $Q$ denote $\mathbb{Z}/2$ as an abelian group and let $X$ be a $Q$-spectrum. If I want to compute the homotopy groups of $X^{hQ}$, homotopy fixed points spectrum, I can use the homotopy fixed points spectral sequence (HFPSS). It's construction is briefly described in Neil's answer to this question: $RO(Q)$-graded homotopy fixed point spectral sequence.

In particular, it's $E_1$ page is given by $E_1^{p,q}=[Q_+\wedge\mathbb{S}^p, X]^{p+q}_Q$. By work of Greenlees and May this is isomorphic to $Hom_{Q}(H_p(Q_+\wedge \mathbb{S}^p), \pi_{-q}(i^*X))$ (I hope signs are correct here). The $d_1$ differential is then induced by the "geometric boundary", i.e. a composition $Q_+\wedge\mathbb{S}^p\to\Sigma\mathbb{S}((p-1)\sigma)_+\to Q_+\wedge\mathbb{S}^{p-1}$. Both maps here are coming from cofiber sequences of the form $\mathbb{S}((k-1)\sigma)_+\to\mathbb{S}(k\sigma)_+\to Q_+\wedge\mathbb{S}^k$. Of course, $H_p(Q_+\wedge\mathbb{S}^p)\cong\mathbb{Z}[Q]$.

Question: Why the map induced by the geometric boundary in $H_p$ is the map arising in the classical $\mathbb{Z}[Q]$-resolution of $\mathbb{Z}$, i.e. either $1+g$ or $1-g$?