Extending the action of quotient group on abelian normal subgroup to bigger abelian group gives rise to canonical bigger group

Statement

Suppose $E$ is a group having an abelian normal subgroup $N$ and a quotient group $B$. Suppose there exists an abelian group $M$ containing $N$ such that the induced action of $B$ on $N$ extends to an action of $B$ on $M$.

Then, there exists a group $F$ containing $E$ and also containing a copy of $M$ containing a copy of $N$, such that $E\cap M=N$ and $EM=F$, and the induced action of $F/M\cong B$ on $M$ is the specified extended action of $B$ on $M$.