Let $R$ be a commutative ring with unity and let $G$ be a finite group. Let $\gamma \in Z^{2}(G,R^{\times})$ be a $2$-cocycle. The *twisted group ring* (of $G$ over $R$ with respect to $\gamma$) $R *_{\gamma} G$ is an $R$-algebra defined as follows:
it is the free $R$-module with basis $\{ u_g \}_{g \in G}$ where the multiplication on the basis is defined via $u_{g}u_{h} = \gamma(g,h)u_{gh}$ for all $g,h \in G$ and $ru_g =u_gr$ for all $g \in G$ and $r \in R$. (Note that this also goes under different names such as *crossed product order*. Moreover, the term *twisted group ring* sometimes has a different but related meaning.)

The structure of $R *_{\gamma} G$ only depends on the class of $\gamma$ in $H^2(G,R^{\times})$. It would appear that this result is well-known to experts (see page 2 of this preprint, for example https://arxiv.org/abs/1611.00499). However, I can't seem to find a proof is this in the literature.

Is there a nice a reference (textbook or article) that gives an introduction to such twisted group rings and in particular proves the above fact on the relation with $H^2(G,R^{\times})$? Ideally, it should be for general $R$ rather than the special case that $R$ is a field.

Note that I don't need a proof (I know how to do this), but rather a reference to a proof, etc.