How can one estimate the number of $p$-groups of order $\leq p^n$ that split over a normal abelian subgroup?
Moreover, let $s(n,p)$ be the number of such groups, and let $f(n,p)$ denotes the number of $p$-groups of order $\leq p^n$.
What can one expect for $\frac{s(n,p)}{f(n,p)}$?
Is it true that $lim_n\frac{s(n,p)}{f(n,p)}$ is 0, for every prime $p$?
