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This paper is a sequel to [He11] and [GH17]. In [He11] a notion of marking ofisolated hypersurface singularities was defined, and a moduli space$M_\mu^{mar}$ for marked singularities in one $\mu$-homotopy class of isolatedhypersurface singularities was established. It is an analogue of aTeichm\"uller space. It comes together with a $\mu$-constant monodromy group$G^{mar}\subset G_{\mathbb{Z}}$. Here $G_{\mathbb{Z}}$ is the group ofautomorphisms of a Milnor lattice which respect the Seifert form. It wasconjectured that $M_\mu^{mar}$ is connected. This is equivalent to $G^{mar}=G_{\mathbb{Z}}$. Also Torelli type conjectures were formulated. In [He11] and[GH17] $M_\mu^{mar}, G_{\mathbb{Z}}$ and $G^{mar}$ were determined and allconjectures were proved for the simple, the unimodal and the exceptionalbimodal singularities. In this paper the quadrangle singularities and thebimodal series are treated. The Torelli type conjectures are true. But theconjecture $G^{mar}= G_{\mathbb{Z}}$ and $M_\mu^{mar}$ connected does not holdfor certain subseries of the bimodal series.

$\mu$-constant monodromy groups and Torelli results for the quadrangle singularities and the bimodal series