Z(2)‐equivariant bifurcation equations in two variables and with a distinguished bifurcation parameter are analysed in the framework of imperfect bifurcation theory in the presence of symmetry. All possible inequivalent bifurcation equations up to codimension 4, together with their universal unfoldings, are collected in a list of normal forms. Conditions are set up which must be satisfied for an arbitrary bifurcation problem to be contact equivalent to a given normal form. The list is supplemented by several normal forms with codimension less than 7 and topological codimension less than or equal to 4.