Enumeration of solvable cube-free groups and counting certain types of split extensions

A group is said to be cube-free if its order is not divisible by the cube of any prime. Let $f_{cf,sol}(n)$ denote the isomorphism classes of solvable cube-free groups of order $n$. We find asymptotic bounds for $f_{cf,sol}(n)$ in this paper. Let $p$ be a prime and let $q=p^k$ for some positive integer $k$. We also give a formula for the number of conjugacy classes of the subgroups that are maximal amongst non-abelian solvable cube-free $p^{\mathrm{\prime }}$-subgroups of $\mathrm{G}\mathrm{L}(2,q)$. Further, we find the exact number of split extensions of $P$ by $Q$ up to isomorphism of a given order where $P\in \{{\mathbb{Z}}_p\times {\mathbb{Z}}_p,{\mathbb{Z}}_{p^{\alpha }}\}$, $p$ is a prime, $\alpha$ is a positive integer and $Q$ is a cube-free abelian group of odd order such that $p\nmid \mathrm{\mid }Q\mathrm{\mid }$.