Enumeration of Groups in Varieties of $A$-groups: A survey

Let $S$ be a class of groups and let $f_S(n)$ be the number of isomorphism classes of groups in $S$ of order $n$. Let $f(n)$ count the number of groups of order $n$ up to isomorphism. The asymptotic bounds for $f(n)$ behave differently when restricted to abelian groups, $A$-groups and groups in general. We survey some results and some open questions in enumeration of finite groups with a focus on enumerating within varieties of $A$-groups.