  The Gamow-Teller transition operator is written as a polynomial in the dipole
proton-neutron and quadrupole charge conserving QRPA boson operators, using the
prescription of the boson expansion technique of Belyaev-Zelevinski type. Then,
the $2\nu\beta\beta$ process ending on the first $2^+$ state in the daughter
nucleus is allowed via one, two and three boson states describing the odd-odd
intermediate nucleus. The approach uses a single particle basis which is
obtained by projecting out the good angular momentum from an orthogonal set of
deformed functions. The basis for mother and daughter nuclei have different
deformations. The GT transition amplitude as well as the half lives were
calculated for ten transitions.
  Results are compared with the available data as well as with some predictions
obtained with other methods.
