On symmetric and twisted-unitary modular units in group rings

We study units in group rings of the Hantzsche-Wendt group. We construct a compatible system of non-trivial unitary units over $\mathbb{Z}\mathrm{/}{2}^n\mathbb{Z}$ for every $n\ge 1$, and ``almost-group’’ units over $\mathbb{Z}\mathrm{/}2\mathbb{Z}$, meaning non-trivial units $u$ with the property that $u-g=u^{-1}-g^{-1}$ for some group element $g$.