  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 $\mathbb Z^2\rtimes\text{SL}_2(\mathbb Z)$ and
$\text{SL}_n(\mathbb Z)$, 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
$\Gamma$ with property (T), including those such that
$\text{H}^2(\Gamma,\mathbb R)\not=0$ or $\text{H}^2(\Gamma,\mathbb
Z\Gamma)\not=0$, do not have the lifting property (LP). More generally, we show
that the same holds if $\Gamma$ admits a probability measure preserving action
with non-vanishing second $\mathbb R$-valued cohomology. Finally, we prove that
the full C$^*$-algebra of any non-finitely presented property (T) group fails
the LP.
