The direct detection of light fermionic Dark Matter (DM) is a technological challenging task. In this work the nuclei that are unstable against the 2$\beta$ decay are considered as a target for the direct detection of a class of light Dark Matter candidates.
In particular, the same diagram responsible for the hypothetical neutrinoless double beta decay could be considered for the detection of a Majorana DM fermion. In this detection approach, the interaction of the DM fermion with the unstable nucleus causes the nucleus decay.
The relatively large energy provided by the nucleus decay provides an experimental signature for this class of light DM fermion.
The expected signal distribution for various DM masses is evaluated and compared with the current data available from the experiments using $^{76}$Ge and $^{136}$Xe target nuclei.
The upper limits on the total cross sections for this inelastic nucleus scattering are shown.