Let $A$ be an abelian group and $G$ be a group. A short exact sequence of groups like $1\longrightarrow A\longrightarrow E\longrightarrow G\longrightarrow 1$ is called an extension. We say that $E$ is an extension of $A$ by $G$. This extension makes $A$ into a $G$-module.

Assume that $A$ is an elementary abelian p-subgroup of rank $m$ and $G$ be an elementary abelian p-subgroup of rank $2$. Then $A$ is a $F_{p}[G]$ -module with $F_{p}$ is a finite field of $p$ elements. Obviously, this structure of $F_{p}[G]$-module is not unique. Thus, one can asks the following question:

Question: What is the number of structure of $F_{p}[G]$-modules defined on $A$?.

Any help would be appreciated so much. Thank you all.

Let $A$ be an abelian group and $G$ be a group. A short exact sequence of groups like $1\longrightarrow A\longrightarrow E\longrightarrow G\longrightarrow 1$ is called an extension. We say that $E$ is an extension of $A$ by $G$. This extension makes $A$ into a $G$-module.

Assume that $A$ is an elementary abelian p-subgroup of rank $m$ and $G$ be an elementary abelian p-subgroup of rank $2$. Then $A$ is a $F_{p}[G]$ -module with $F_{p}$ is a finite field of $p$ elements. Obviously, this structure of $F_{p}[G]$-module is not unique. Thus, one can ask the following question:

Question: What is the number of structure of $F_{p}[G]$-modules defined on $A$?.

Any help would be appreciated so much. Thank you all.