This is the second to three talks on the non-existence of certain Galois extensions of the rational number field Q with prescribed ramification, motivated by Serre's conjecture. This talk will present the paper of Sharon Brueggeman on the non-existence of certain continuous irreducible mod p representations of degree 2 of the absolute Galois group of the rational number field in the case p=5, and will also present a generalization to other primes.