The question does not explicitly ask for online tools and online resources. Still, I suspect that many answers might be about something available online.
@FrançoisG.Dorais I see that Alexander Chervov did not ping you. So I will add ping instead of him - just to make sure that you do not miss his response here in chat: chat.stackexchange.com/transcript/message/40581610#40581610 (Continuation of the open-problems-list discussion.)
Let $\mathbb{Z}^\text{inv}$ denote the $\mathbb{Z}/2$-module defined by the inversion action on $\mathbb{Z}$. Let $G_0$ be a finite group that acts trivially on $\mathbb{Z}^\text{inv}$. Then one can compute via the Künneth Theorem and UCT for cohomology,\begin{align}H^3(G_0\times\mathbb{Z}/2;\mat...
