The topological classification of gerbes, as principal bundles with the structure group the projective unitary group of a complex Hilbert space, over a topological space H is given by the third cohomology H-3 (H, Z). When H is a topological group the integral cohomology is often related to a locally continuous (or in the case of a Lie group, locally smooth) third group cohomology of H. We shall study in more detail this relation in the case of a group extension 1 -> N -> G -> H -> 1 when the gerbe is defined by an abelian extension 1 -> A -> N -> 1 of N. In particular, when H-s(1) (N, A) vanishes we shall construct a transgression map H-s(2) (N, A) -> H-s(3) (H, A(N)), where A(N) is the subgroup of N-invariants in A and the subscript s denotes the locally smooth cohomology. Examples of this relation appear in gauge theory which are discussed in the paper. (C) 2016 Elsevier B.V. All rights reserved.