Given a Jordan domain Ω in the extended complex plane C, denote byMb(Ω), M(Ω) and R(Ω) the boundary quasiextremal distance constant, quasiextremal dis-tance constant and quasiconformal reflection constantofS2.respectively. It is known thatMb(Ω)<M(Ω)<R(Ω)+1. In this paper, we will give some further relations amongMb (Ω), M(Ω) and R (Ω) by introducing and studying some other closely related constants.Particularly, we will give a necessary and sufficient condition for Mb(Ω) = R(Ω)+1 andshow that M(Ω)<R (Ω)+I for all asymptoticajly conformal extension domains other thandisks. This gives an affirmative answer to a question asked by Yang, showing that the con jecture M(Ω)=R(Ω)+I by Garnett and Yang is not true for all asymptoticallyconformalextension domains other than disks. Our discussion relies heavily on the theory of extremalquasiconformal mappings. which in turn gives some interesting results in the extremal quasi-conformal mapping theory as well.