A question of Frohardt on $2$-groups, and skew translation quadrangles of even order

We solve a fundamental question posed in Frohardts 1988 paper [Fro] on finite $2$-groups with Kantor familes, by showing that finite groups with a Kantor family $(mathcal{F},mathcal{F}^*)$ having distinct members $A, B in mathcal{F}$ such that $A^* cap B^*$ is a central subgroup of $H$ and the quotient $H/(A^* cap B^*)$ is abelian cannot exist if the center of $H$ has exponent $4$ and the members of $mathcal{F}$ are elementary abelian. In a similar way, we solve another old problem dating back to the 1970s by showing that finite skew translation quadrangles of even order $(t,t)$ are always translation generalized quadrangles.