using a 2-cocycle over $G^{\vee}$.

#### Construction 2 {#subsubsec: constr2}

We now move on to the second construction. One first defines an action of $G$ on the twisted group algebra $kG_{\mu}$ by $k$-algebra automorphisms. The algebra $kG_{\mu}$ can be obtained by taking $A=k$ in Example \[ex: twistedgroupalgebra\].

The action of $G$ on $kG_{\mu}$ is defined as follows: for all $g,h \in G$ and $\alpha \in k$, let $(\alpha \otimes_k
g)^h:=\chi_g(h) \alpha\otimes_k g$. By assumption one has an action of $G$ on $A$ by $k$-algebra automorphisms, thus one can define a diagonal action of $G$ on the tensor product $A \otimes_k kG_{\mu}$ by $$\label{eq: diagactionconstr2}
(a \otimes_k g)^h := \chi_g(h)(a^h \otimes_k g),$$ for all $a \in A$ and $g, h \in G$.

The construction is then completed by taking the invariant ring under this action, $(A \otimes_k kG_{\mu})^G$, which is isomorphic to $(AG_{\mu})^G$ as a $k$-algebra.

Note that for the Klein-four group $G$ and a particular normalised 2-cocycle $\mu$ one has $kG_{\mu} \cong M_2(k)$ as $k$-algebras by Lemma \[lem: kgmuiso\]. This provides some indication that Example \[ex: odesskii\] can be described using Construction 2. The demonstration of this is delayed until §\[subsec: odesskiiegtwist\].

#### Consequences {#subsubsec: consequences}

We now gather together 