Transitive $(q-1)$-fold packings of ${\mathrm{P}\mathrm{G}}_n(q)$

A $t$-fold packing of a projective space ${\mathrm{P}\mathrm{G}}_n(q)$ is a collection $\mathcal{P}$ of line-spreads such that each line of ${\mathrm{P}\mathrm{G}}_n(q)$ occurs in precisely $t$ spreads in $\mathcal{P}$. A $t$-fold packing $\mathcal{P}$ is transitive if a subgroup of ${\mathrm{P}\mathrm{\Gamma }\mathrm{L}}_{n+1}(q)$ preserves and acts transitively on $\mathcal{P}$. We give a construction for a transitive $(q-1)$-fold packing of ${\mathrm{P}\mathrm{G}}_n(q)$, where $q=2^k$, for any odd positive integers $n$ and $k$, such that $n\ge 3$. This generalises a construction of Baker from 1976 for the case $q=2$.