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Within the PHFB approach, the $0^{+}\rightarrow 2^{+}$ transition of twoneutrino double-$\beta $ decay of $^{94,96}$Zr, $^{100}$Mo, $^{104}$Ru,$^{110}$Pd, $^{128,130}$Te and $^{150}$Nd isotopes is studied employing wavefunctions generated with four different parametrizations of the pairing plusmultipole type of two-nucleon interaction and the summation method. Incomparison to the $0^{+}\rightarrow 0^{+}$ transition, the nuclear transitionmatrix elements $M_{2\nu }(2^{+})$ are quite sensitive to the deformation ofthe yrast 2$^{+}$ state. Consideration of the available theoretical andexperimental results suggest that the observation of the 0$% ^{+}\rightarrow$2$^{+}$ transition of $2\nu \beta ^{-}\beta ^{-}$ decay may be possible in$^{96}$Zr, $^{100}$Mo, $^{130}$Te and $^{150}$Nd isotopes. The effect ofdeformation on the $M_{2\nu }(2^{+})$ is also studied.

Two neutrino double-$\beta $ decay of $94\leq A\leq 150$ nuclei for the 0$^{+}\rightarrow $2$^{+}$ transition