The two-step ageing of Fe-28Ni-17Co-11.5Al-2.5Nb (at. %) single crystals under and without stress, leads to the precipitation of the γ′- and β-phase particles. Research has shown that γ–α′ thermoelastic martensitic transformation (MT), with shape memory effect (SME) and superelasticity (SE), develops in the [001]-oriented crystals under tension. SE was observed within the range from the temperature of the start of MT upon cooling Ms, to the temperature of the end of the reverse MT upon heating Af, and at temperatures from Af to 323–373 K. It was found that at γ–α′ MT in the [001]-oriented crystals, with γ′- and β-phase particles, a high level of elastic energy, ΔGel, is generated, which significantly exceeds the energy dissipation, ΔGdis. As a result, the temperature of the start of the reverse MT, while heating As, became lower than the temperature Ms. The development of γ–α′ MT under stress occurs with high values of the transformation hardening coefficient, Θ = dσ/dε from 2 to 8 GPa and low values of mechanical Δσ and thermal ΔTh hysteresis. The reasons for an increase in ΔGel during the development of γ–α′ MT under stress are discussed.