Let α :Q→ Q be an involution of a Hilbert cube with fixed point set Q^α that has Property Z in Q. The first main result of this paper is Theorem 3.1: Assume that (Q,α) is an absolute retract in the category of metric spaces with involutions and equivariant maps. If T⊆ Q is an equivariant retract of Q containing Q^α that is an inequivariant Z-set in Q, then for any equivariant retraction r:Q→ T, Q is equivariantly homeomorphic with the mapping cylinder M(r;T) of r reduced at T. The second main result is part of Theorem 3.3: Q^α is an equivariant strong deformation retract of Q if and only if Q is equivariantly homeomorphic with Q^α× Π_i≧1I_i equipped with the involution that reflects each interval coordinate I_i across its mid-point.