The theories of interest include General Relativity formulated in terms of tetrads, Einstein-Yang-Mills theory and Einstein-Dirac theory. For these theories having internal gauge freedom, we are going to see properties of their natural group action of diffeomorphisms of spacetime on such dynamical fields. For Einstein-Dirac equations, we go through several attacked and under-attacked situations of searching its solutions. We hope to figure out the horizon thermodynamics for Einstein-Dirac theory. We further want to investigate the Einstein-Dirac theory with aether field to see if we can have the Einstein-Dirac-aether solutions.

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