Suppose we have an abelian extension

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

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 $A=k^G\times kF$ as algebras.

Is it true that $\sigma^n=\mathrm{id}$ for some $n\geq 1$? In other words 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. 

Another question, is the set of all cocycles ("Schur multiplier" ) with values in $k^G$ finite.