  Let A be the adele ring over a totally real number field F. For cohomological
cuspidal automorphic irreducible representations of GSp(4,A) coming from weak
endoscopic or Saito-Kurokawa Lifts we determine the local invariant spaces
under the first principal congruence subgroup at the non-archimedean places.
For F=Q this gives rise to dimension formulas regarding certain subspaces of
the inner cohomology of the genus two Shimura variety corresponding to the
principal congruence subgroup level N=2. We prove the conjectures made by
Bergstr\"om, Faber and van der Geer in a recent paper.
