  Fermionic extensions of generic 2d gravity theories obtained from the graded
Poisson-Sigma model (gPSM) approach show a large degree of ambiguity. In
addition, obstructions may reduce the allowed range of fields as given by the
bosonic theory, or even prohibit any extension in certain cases. In our present
work we relate the finite W-algebras inherent in the gPSM algebra of
constraints to algebras which can be interpreted as supergravities in the usual
sense (Neuveu-Schwarz or Ramond algebras resp.), deformed by the presence of
the dilaton field. With very straightforward and natural assumptions on them
--like demanding rigid supersymmetry in a certain flat limit, or linking the
anti-commutator of certain fermionic charges to the Hamiltonian constraint-- in
the ``genuine'' supergravity obtained in this way the ambiguities disappear, as
well as the obstructions referred to above. Thus all especially interesting
bosonic models (spherically reduced gravity, the Jackiw-Teitelboim model etc.)\
under these conditions possess a unique fermionic extension and are free from
new singularities. The superspace supergravity model of Howe is found as a
special case of this supergravity action. For this class of models the relation
between bosonic potential and prepotential does not introduce obstructions as
well.
