For the cohomology groups of locally symmetric spaces attached to a connected reductive group $G$ we developed a topological trace formula describing the action of Hecke operators twisted by an outer automorphism $\eta$. In some cases this twisted trace formula is stable and may be compared with the untwisted trace formula for the stable endoscopic group $G_1$. The fundamental lemma for this comparison in the case $G=PGl_5$, $G_1=Sp_4$ can be shown (joint work with J. Ballmann and R. Weissauer) to be equivalent to the fundamental lemma in the case $G=GL_4\times GL_1$ and $G_1=GSp_4=GSpin_5$ (proved by Flicker). Using character identities between local representations and properties of the $\theta$-lift one gets multiplicity results for automorphic representations contributing to the cohomology of Siegel modular 3-folds.