Isomorphisms of quadratic quasigroups

Let $\mathbb{F}$ be a finite field of odd order and $a,b\in \mathbb{F}\setminus \{0,1\}$ be such that $\chi (a)=\chi (b)$ and $\chi (1-a)=\chi (1-b)$, where $\chi$ is the extended quadratic character. Let $Q_{a,b}$ be the quasigroup upon $\mathbb{F}$ defined by $(x,y)\mapsto x+a(y-x)$ if $\chi (y-x)\ge 0$, and $(x,y)\mapsto x+b(y-x)$ if $\chi (y-x)=-1$. We show that $Q_{a,b}\cong Q_{c,d}$ if and only if $\{a,b\}=\{\alpha (c),\alpha (d)\}$ for some $\alpha \in \text{aut}(\mathbb{F})$. We also characterise $\text{aut}(Q_{a,b})$ and exhibit further properties, including establishing when $Q_{a,b}$ is a Steiner quasigroup or is commutative, entropic, left or right distributive, flexible or semisymmetric. In proving our results we also characterise the minimal subquasigroups of $Q_{a,b}$.