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In this paper, we construct a Real equivariant version of the B\"okstedt spectral sequence which takes inputs in the theory of Real Hochschild homology developed by Angelini-Knoll, Gerhardt, and Hill and converges to the equivariant homology of Real topological Hochschild homology, THR. We also show that when A is a commutative C₂-ring spectrum, THR (A) has the structure of an A-Hopf algebroid in the C₂-equivariant stable homotopy category.
C. I. Lewis (Tue,) studied this question.