I am struggling with exercise 3.9.6 presented in Clifford Bergman's book "Universal Algebra. Fundamentals and Selected Topics". The exercise is as following:
Let $V$ and $W$ be varieties of groups. Define $V \cdot W$ to be the class of all groups $A$ containing a normal subgroup $B$ such that $B \in V$ and $A/B \in W$. Prove that $V \cdot W$ is a variety.
I strongly suspect that this task requires from us using the algebraic isomorphisms theorems but I have no idea how to start.
Thank you in advance.