Suppose we have an abelian extension

$$k \rightarrow k^G \rightarrow A \rightarrow kF \rightarrow k.$$

According to the general theory there is a left action of $F$ on $G$ and a $2$-cocycle $\sigma:F\times F \rightarrow k^G$ such that $A=k^G\#_{\sigma}kF$ as algebras.

Is it true that $\sigma^n=\mathrm{id}$ for some $n\geq 1$? In other words does it follow that $\sigma_a(f,h)^n=\mathrm{id}$ if $\sigma(f, h)=\sum_{a \in G}\sigma_a(f,h)p_a$ where $p_a$ is the usual notation for dual basis of the group element basis.

A related question, is the set of all $2$-cocycles of $F$ with values in $k^G$ finite.