  Here, we study the neutrinoless double-$\beta$ ($0\nu\beta\beta$) decay
between 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 the
microscopic interacting boson model (IBM-2). The IBM-2 has been widely used to
obtain predictions for nuclear observables, such as the spectrum, but also to
explore the possible emergence of beyond-the-Standard Model effects in the weak
interactions of nuclei. Our calculations are carried out by considering the
exchange of a Majorana neutrino between two nucleons ($2N$-mechanism). In
addition to NMEs, we calculate the associated leptonic phase-space factors
(PSFs) using electron radial wave functions, which are obtained by solving
numerically the Dirac equation of a screened Coulomb potential that takes into
account finite nuclear size. By combining our IBM-2 results for the NMEs with
those for the PSFs along with experimental half-life limits, we can set limits
on the $\langle \lambda \rangle$ and $\langle \eta \rangle$ couplings of
left-right (L-R) models.
