 Counting Sylow bases in a coprime elementary abelian extension

Let $V$ be an elementary abelian $r$-group and let $r\nmid |H|$, with $H$ soluble. In each Sylow basis of $V⋊H$, $V$ is the unique Sylow $r$-subgroup and the product of the remaining Sylow subgroups is a complement to $V$. All complements are $V$-conjugate, and the stabilizer of $H$ under this conjugation is its fixed-vector space $C_V(H)$. Each complement has $b(H)$ Sylow bases, proving the formula. For the permutation module $V={\mathbb{F}}_5^4$ of $S_4$, the fixed-vector space is the one-dimensional constant-coordinate subspace, so the number is $5^3\cdot 12=1500$.