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Here, we study the neutrinoless double-$\beta$ ($0\nu\beta\beta$) decaybetween the ground state and the first $2^+$ state of $^{76}\mbox{Ge}\rightarrow {}^{76}\mbox{Se}$, $^{82}\mbox{Se} \rightarrow{}^{82}\mbox{Kr}$,$^{130}\mbox{Te} \rightarrow {}^{130}\mbox{Xe}$ and $^{136}\mbox{Xe}\rightarrow {}^{136}\mbox{Ba}$ systems. The relevant nuclear matrix elements(NMEs) involved in the process are calculated within the formalism of themicroscopic interacting boson model (IBM-2). The IBM-2 has been widely used toobtain predictions for nuclear observables, such as the spectrum, but also toexplore the possible emergence of beyond-the-Standard Model effects in the weakinteractions of nuclei. Our calculations are carried out by considering theexchange of a Majorana neutrino between two nucleons ($2N$-mechanism). Inaddition to NMEs, we calculate the associated leptonic phase-space factors(PSFs) using electron radial wave functions, which are obtained by solvingnumerically the Dirac equation of a screened Coulomb potential that takes intoaccount finite nuclear size. By combining our IBM-2 results for the NMEs withthose for the PSFs along with experimental half-life limits, we can set limitson the $\langle \lambda \rangle$ and $\langle \eta \rangle$ couplings ofleft-right (L-R) models.

$0^+$ to $2^+$ neutrinoless double-$\beta$ decay of $^{76}$Ge, $^{82}$Se, $^{130}$Te and $^{136}$Xe in the microscopic interacting boson model}