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Let $G$ be a finite group; denote by $\mathbb{Z}_2$ the cyclic group of order $2$. Let $\pi: G \rightarrow \mathbb{Z}_2$ be a non-trivial group homomorphism. Let M be the $G$ representation $\mathbb{Z}_2 \times \mathbb{Z}_2$ with action given by \begin{align} g[(a,b)] = \begin{cases} (a,b)& \quad \text{if $\pi(g) = 0$}\\ (b,a)& \quad \text{if $\pi(g) = 1$}. \end{cases} \end{align}

By sending a cocycle $t \in Z^2(G,\mathbb{Z}_2)$ to $(t,t) \in Z^2(G,M)$ we see that $Z^2(G,\mathbb{Z}_2)$ is a subgroup of $Z^2(G,M)$.

What's an example of a group $G$ and normalized cocycle in $Z^2(G,M)$ which is not in the subgroup $Z^2(G,\mathbb{Z}_2)$?

Let $G$ be a finite group; denote by $\mathbb{Z}_2$ the cyclic group of order $2$. Let $\pi: G \rightarrow \mathbb{Z}_2$ be a non-trivial group homomorphism. Let M be the $G$ representation $\mathbb{Z}_2 \times \mathbb{Z}_2$ with action given by \begin{align} g[(a,b)] = \begin{cases} (a,b)& \quad \text{if $\pi(g) = 0$}\\ (b,a)& \quad \text{if $\pi(g) = 1$}. \end{cases} \end{align}

Let \begin{align} f: M &\rightarrow \mathbb{Z}_2\\ f(a,b) & \mapsto a+b \end{align}

$f$ maps a 2-cocycle in $Z^2(G,M)$ to a 2-coycle in $Z^2(G,\mathbb{Z}_2)$. I want an example where the image of $f$ is not a coboundary.

Examples I have checked:

$H^2(\mathbb{Z}_2, M) = 0$ for $\pi = \text{id}$.

I have also checked numerically that $H^2(\mathbb{Z}_2 \times \mathbb{Z}_2, M)$ for $\pi(a,b) = a+b$ and $\pi(a,b)=a$ also only have images that are coboundaries under $f$.

Let $G$ be a finite group; denote by $\mathbb{Z}_2$ the cyclic group of order $2$. Let $\pi: G \rightarrow \mathbb{Z}_2$ be a non-trivial group homomorphism. Let M be the $G$ representation $\mathbb{Z}_2 \times \mathbb{Z}_2$ with action given by \begin{align} g[(a,b)] = \begin{cases} (a,b)& \quad \text{if $\pi(g) = 0$}\\ (b,a)& \quad \text{if $\pi(g) = 1$}. \end{cases} \end{align}

By sending a cocycle $t \in Z^2(G,\mathbb{Z}_2)$ to $(t,t) \in Z^2(G,M)$ we see that $Z^2(G,\mathbb{Z}_2)$ is a subgroup of $Z^2(G,M)$.

When is $Z^2(G,M)/Z^2(G,\mathbb{Z}_2)$ non-trivial?

What's an example of a normalized cocycle in $Z^2(G,M)$ which is not in the subgroup $Z^2(G,\mathbb{Z}_2)$?

Let $G$ be a finite group; denote by $\mathbb{Z}_2$ the cyclic group of order $2$. Let $\pi: G \rightarrow \mathbb{Z}_2$ be a non-trivial group homomorphism. Let M be the $G$ representation $\mathbb{Z}_2 \times \mathbb{Z}_2$ with action given by \begin{align} g[(a,b)] = \begin{cases} (a,b)& \quad \text{if $\pi(g) = 0$}\\ (b,a)& \quad \text{if $\pi(g) = 1$}. \end{cases} \end{align}

By sending a cocycle $t \in Z^2(G,\mathbb{Z}_2)$ to $(t,t) \in Z^2(G,M)$ we see that $Z^2(G,\mathbb{Z}_2)$ is a subgroup of $Z^2(G,M)$.

What's an example of a group $G$ and normalized cocycle in $Z^2(G,M)$ which is not in the subgroup $Z^2(G,\mathbb{Z}_2)$?

Let $G$ be a finite group; denote by $\mathbb{Z}_2$ the cyclic group of order $2$. Let $\pi: G \rightarrow \mathbb{Z}_2$ be a non-trivial group homomorphism. Let M be the $G$ representation $\mathbb{Z}_2 \times \mathbb{Z}_2$ with action given by \begin{align} g[(a,b)] = \begin{cases} (a,b)& \quad \text{if $\pi(g) = 0$}\\ (b,a)& \quad \text{if $\pi(g) = 1$}. \end{cases} \end{align}

By sending a cocycle $t \in Z^2(G,\mathbb{Z}_2)$ to $(t,t) \in H^2(G,M)$ we see that $Z^2(G,\mathbb{Z}_2)$ is a subgroup of $H^2(G,M)$.

When is $H^2(G,M)/Z^2(G,\mathbb{Z}_2)$ non-trivial?

What's an example of a normalized cocycle in $H^2(G,M)$ which is not in the subgroup $Z^2(G,\mathbb{Z}_2)$?

Let $G$ be a finite group; denote by $\mathbb{Z}_2$ the cyclic group of order $2$. Let $\pi: G \rightarrow \mathbb{Z}_2$ be a non-trivial group homomorphism. Let M be the $G$ representation $\mathbb{Z}_2 \times \mathbb{Z}_2$ with action given by \begin{align} g[(a,b)] = \begin{cases} (a,b)& \quad \text{if $\pi(g) = 0$}\\ (b,a)& \quad \text{if $\pi(g) = 1$}. \end{cases} \end{align}

By sending a cocycle $t \in Z^2(G,\mathbb{Z}_2)$ to $(t,t) \in Z^2(G,M)$ we see that $Z^2(G,\mathbb{Z}_2)$ is a subgroup of $Z^2(G,M)$.

When is $Z^2(G,M)/Z^2(G,\mathbb{Z}_2)$ non-trivial?

What's an example of a normalized cocycle in $Z^2(G,M)$ which is not in the subgroup $Z^2(G,\mathbb{Z}_2)$?