Point configurations on finite dimensional real or complex spaces, typically on the unit sphere, are important in Physics, Coding Theory, Classical and Quantum Information Theory, Geometry, Number Theory and more. The Automorphism group a of point configuration is a tool to study it, and to generate new ones. In this talk we will show how to generate automorphism groups from group-theoretic considerations, and how to construct configurations that satisfy the group. The main tool in use is Group Cohomology. We show that there is a spectral sequence which captures all possible solutions and all obstructions to the construction of a solution. In addition this sequence captures the Galois structure of algebraic configurations. Galois structures were discovered recently in the case of Zauner SIC-POVMs. This point of view can be generalized to a much broader framework, e.g. higher tensors as replacements of the Gramian matrix, perfect squares are seen to be dual to configuration Gramians when one uses homology instead of cohomology, and there are some connections to Number Theory, such as the theory of Brauer Groups. Other applications are the generation of Hadamard and Weighing matrices. We will discuss these extensions as time permits. This is joint work with Giora Dula.