In order to study the running coupling in four-flavour QCD, we review the set-up of the Schrödinger functional (SF) with staggered quarks. Staggered quarks require lattices which, in the usual counting, have even spatial lattice extent L/a while the time extent T /a must be odd. Set- ting T = L is therefore only possible up to O(a), which introduces different cutoff effects already in the pure gauge theory. We re-define the SF such as to cope with this situation and determine the corresponding classical background field. A perturbative calculation yields the coefficient of the pure gauge O(a) boundary counterterm to one-loop order.