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Examples of a group $G$ and an $F$-representation $V$ where $\cup:H^1(G,F)\otimes H^1(G,V)\t...
$\newcommand{\bZ}{\mathbb{Z}}$Let $G$ be a finite group of order divisible by $p:=\mathrm{char}\, F$ such that $\dim_F \mathrm{Hom}(G,F)\geq 2$ (e.g. $G=\bZ/p\times\bZ/p$). Take $V$ to be the kernel o …