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We generalize the standard $N=2$ supersymmetric Kazama-Suzuki cosetconstruction to the $N=4$ case by requiring the {\it non-linear}(Goddard-Schwimmer) $N=4~$ quasi-superconformal algebra to be realized oncosets. The constraints that we find allow very simple geometricalinterpretation and have the Wolf spaces as their natural solutions. Our resultsobtained by using components-level superconformal field theory methods arefully consistent with standard results about $N=4$ supersymmetrictwo-dimensional non-linear sigma-models and $N=4$ WZNW models on Wolf spaces.We construct the actions for the latter and express the quaternionic structure,appearing in the $N=4$ coset solution, in terms of the symplectic structureassociated with the underlying Freudenthal triple system. Next, we gauge the$N=4~$ QSCA and build a quantum BRST charge for the $N=4$ string propagating ona Wolf space. Surprisingly, the BRST charge nilpotency conditions rule out thenon-trivial Wolf spaces as consistent string backgrounds.Comment: 31 pages, LaTeX, special macros are include

NoN=4 strings on Wolf spaces