A homeomorphismfof a setfof a setis said to be anL-quasiisometry offor alland ifwhenever ∞ ∈E. Nextfis said to have anM-quasiisometric extension toF⊂R2if there exists anM-quasiisometrygofE∪Fsuch that g|E=fSuppose next thatDis a Jordan domain inR2with boundaryCand that ϕ is anL-quasiisometry ofC.This paper contains complete solutions to the following Schoenflies type extension problems.Problem A.Given a domainDfor which quasiisometries ϕ ofCdoes ϕ have a quasiisometric extensiongtoD? Problem B.For which domainsDdoes each quasiisometry ϕ ofChave a quasiisometric extensiongtoD? It turns out that whenC⊂R2, each quasiisometry ϕ ofChas a quasiisometric extension toDif and only ifDis a quasidisk.