Galois representations modulo p that do not lift modulo p^2

For every finite group H and every finite H-module A, we determine the subgroup of negligible classes in H^2(H,A), in the sense of Serre, over fields with enough roots of unity. As a consequence, we show that for every odd prime p and every field F containing a primitive p-th root of unity, there exists a continuous 3-dimensional mod p representation of the absolute Galois group of F(x_1,...,x_p) which does not lift modulo p^2. We also construct continuous 5-dimensional Galois representations mod 2 which do not lift modulo 4. This answers a question of Khare and Serre, and disproves a conjecture of Florence. This is joint work with Alexander Merkurjev.