We develop a new method, based on non-vanishing of second cohomology groups, for proving the failure of lifting properties for full C*-algebras of countable groups with (relative) property (T). We derive that the full C*-algebras of the groups $\mathbb Z^2\rtimes \mathrm{SL}_2(\mathbb Z)$ and $\mathrm{SL}_n(\mathbb Z)$, for $n\geq 3$, do not have the local lifting property (LLP). We also prove that the full C*-algebras of a large class of groups with property (T) do not have the lifting property (LP). This is based on joint work with A. Ioana and P. Spaas.