5 Rephrased a little the question; added 24 characters in body; added 2 characters in body

Suppose we have an abelian extension of Hopf algebras,

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

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

1)Is it true that $\sigma^n=\mathrm{id}$ for some $n\geq 1$? In other words does it follow that can one choose $\sigma_a(f,h)^n=\mathrm{id}$ \sigma'$in the same class with$\sigma$such that if$\sigma(f, \sigma'(f, h)=\sum_{a \in G}\sigma_a(f,h)p_a$where G}\sigma'_a(f,h)p_a$ then $$\sigma\;'_a(f,h)^n=\mathrm{id}$$ for all $a, f, h$. Here $p_a$ is the usual notation for dual basis of the group element basis .

2) A related little more general question , (that implies the first question) is if the set of all $2$-cocycles of $F$ with values in $k^G$ finite? Equivalently the question is if $H^2(F, k^G)$ is finite with $F$ acting on $k^G$ via the action of $F$ on $G$.

3)A related question for what groups $X$ with an $F$-action the group $H^2(F, X)$ is finite? For example $X=k^*$ gives the usual Schur multiplier.

4 Fixed LaTeX and added clarifying point about the extension from OP's comment

Suppose we have an abelian extension of Hopf algebras,

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

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 $A=k^G#_{\sigma}kF$ 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.? 3 added 1 characters in body Suppose we have an abelian extension $$k \rightarrow k^G \rightarrow A \rightarrow F kF \rightarrow k.$$ 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.