Units in group rings and blocks of Klein four or dihedral defect

We obtain restrictions on units of even order in the integral group ring $\mathbb{Z}G$ of a finite group $G$ by studying their actions on the reductions modulo $4$ of lattices over the $2$-adic group ring ${\mathbb{Z}}_{2}G$. This improves the “lattice method” which considers reductions modulo primes $p$, but is of limited use for $p=2$ essentially due to the fact that $1\equiv -1(\text{mod }2)$. Our methods yield results in cases where ${\mathbb{Z}}_{2}G$ has blocks whose defect groups are Klein four groups or dihedral groups of order $8$. This allows us to disprove the existence of units of order $2p$ for almost simple groups with socle $\mathrm{PSL}(2,p^f)$ where $p^f\equiv \pm 3(\text{mod }8)$ and to answer the Prime Graph Question affirmatively for many such groups.