Quantifying Quillen's Uniform ${\mathcal{F}}_p$-isomorphism Theorem

Let $G$ be a finite group with $2$-Sylow subgroup of order less than or equal to 16. For such a $G$, we prove a quantified version of Quillen’s uniform ${\mathcal{F}}_p$-isomorphism theorem, which holds uniformly for all $G$-spaces. We do this by bounding from above the exponent of Borel equivariant ${\mathbf{F}}_2$-cohomology, as introduced by Mathew-Naumann-Noel, with respect to the family of elementary abelian 2-subgroups.