For Schur representations in Azumaya algebras, the coarse moduli space is a quotient algebraic space, and the stack of twisted representations is equivalent to the corresponding quotient stack.
Artin: Th´ eor` emes de repr´ esentabilit´ e pour les espaces alg´ ebriques
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Rational points in coarse moduli spaces and twisted representations
For Schur representations in Azumaya algebras, the coarse moduli space is a quotient algebraic space, and the stack of twisted representations is equivalent to the corresponding quotient stack.