  The proton-neutron monopole Lipkin model, which exhibits some properties
which are relevant for those double beta decay ($\beta \beta$) transitions
mediated by the Fermi matrix elements, is solved exactly in the proton-neutron
two-quasiparticle space. The exact results are compared with the ones obtained
by using the Quasiparticle Random Phase (QRPA) and renormalized QRPA (RQRPA)
approaches. It is shown that the RQRPA violates the Ikeda Sum Rule and that
this violation may be common to any extension of the QRPA where scattering
terms are neglected in the participant one-body operators as well as in the
Hamiltonian. This finding underlines the need of additional developments before
the RQRPA could be adopted as a reliable tool to compute $\beta \beta$
processes.
