This paper analyses the steady-state bifurcation with icosahedral symmetry. The Equivariant Branching Lemma is used to predict the generic bifurcating solution branches corresponding to each irreducible representation of the icosahedral group . The relevant amplitude equations are deduced from the equivariance condition, and used to investigate the stability of bifurcating solutions. It is found that the bifurcation with icosahedral symmetry can lead to competition between two-fold, three-fold and five-fold symmetric structures, and between solutions with tetrahedral, three-fold and two-fold symmetry. Stable heteroclinic cycles between solutions with symmetry are found to exist in one of the irreps. The theoretical scenarios are compared with the observed behaviour of icosahedral viruses and nanoclusters.