  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.
