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We continue our investigation of the connected components of the moduli spaceof surfaces of general type containing the Burniat surfaces, correcting amistake in part II. We define the family of extended Burniat surfaces withK_S^2 = 4, resp. 3, and prove that they are a deformation of the family ofnodal Burniat surfaces with K_S^2 = 4, resp. 3. We show that the extendedBurniat surfaces together with the nodal Burniat surfaces with K_S^2=4 form aconnected component of the moduli space. We prove that the extended Burniatsurfaces together with the nodal Burniat surfaces with K_S^2=3 form anirreducible open set in the moduli space. Finally we point out an interestingpathology of the moduli space of surfaces of general type given together with agroup of automorphisms G. In fact, we show that for the minimal model S of anodal Burniat surface (G = (\ZZ/2 \ZZ)^2) we have Def(S,G) \neq Def(S), whereasfor the canonical model X it holds Def(X,G) = Def(X). All deformations of Shave a G-action, but there are different deformation types for the pairs (S,G)of the minimal models S together with the G-action, while the pairs (X,G) havea unique deformation type.

Burniat surfaces III: deformations of automorphisms and extended Burniat surfaces