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 $$M$$.