  In this paper, we consider an equivariant Hopf bifurcation of relative
periodic solutions from relative equilibria in systems of functional
differential equations respecting $\Gamma \times S^1$-spatial symmetries. The
existence of branches of relative periodic solutions together with their
symmetric classification is established using the equivariant twisted
$\Gamma\times S^1$-degree with one free parameter. As a case study, we consider
a delay differential model of coupled identical passively mode-locked
semiconductor lasers with the dihedral symmetry group $\Gamma=D_8$.
