This question is set on a finite $2$-group $G$ and a subgroup $H$ of index $2$ (but perhaps the question could be answered for arbitrary orders/indexes).
A while ago someone posted a question on this site, asking whether $|Ker(res^G_H)|\le 2\cdot|H_2(G,\mathbb{Z})|$, where $res^G_H:H_1(G,\mathbb{Z})\rightarrow H_1(H,\mathbb{Z})$ is the restriction map in dimension $1$ (also known as the Verlagerung transfer map $Ver$). Using the integral coefficients module $\tilde{\mathbb{Z}}$ with the twisted $G$-action $g\cdot z=-z$ for $g\notin H$, I was able to establish the relation $|Ker(res^G_H)|\le |H_2(G,\tilde{\mathbb{Z}})|$.
Question: Is there any feasible way to obtain $H_2(G,\tilde{\mathbb{Z}})$ from $H_2(G,\mathbb{Z})$, or even just relations of orders?
There are the long exact sequences $\cdots\rightarrow H_2(G,\tilde{\mathbb{Z}})\stackrel{\delta}{\rightarrow} H_1(G)\stackrel{res}{\rightarrow}H_1(H)\rightarrow H_1(G,\tilde{\mathbb{Z}})$ and $\cdots\rightarrow H_2(G,\tilde{\mathbb{Z}})\rightarrow H_2(H)\stackrel{cor}{\rightarrow} H_2(G)\stackrel{\partial}{\rightarrow} H_1(G,\tilde{\mathbb{Z}})$ which arise from the short exact sequences of modules $\mathbb{Z}\hookrightarrow Ind^G_H\mathbb{Z}\twoheadrightarrow\mathbb{Z}$ with the twisted-action on either the first or last $\mathbb{Z}$, respectively. But this ultimately leaves you with a relation based on images and kernels that doesn't give sufficient information. There is also the Serre spectral sequence $E_2^{p,q}=H_p(G/H,H_q(H,\tilde{\mathbb{Z}}))\Rightarrow H_{p+q}(G,\tilde{\mathbb{Z}})$, but it's too tough to crack open, and doesn't use the untwisted-homologies of $G$.