We introduce a topological degree theory for the study of Hopf bifurcations in coupled cell systems whose quotient systems (obtained by restricting the system to its flow-invariant subspaces) possess various symmetries. To describe the structure of these quotient symmetries, we introduce the concept of a representation lattice, which is defined as a lattice of representation spaces of (different) symmetry groups that satisfy a compatibility and a consistence condition. Based on the (twisted) equivariant degree, we define a lattice-equivariant degree for maps that are compatible with respect to this representation lattice structure. We apply the lattice-equivariant degree to study a synchrony-breaking Hopf-bifurcation problem in (homogeneous) coupled cell systems and obtain a topological classification of all bifurcating branches of oscillating solutions according to their synchrony types and their symmetric properties.