Michael Weiss considers in his paper "Hammock localization in Waldhausen categories" the Waldhausen category of $G$-CW-spectra ($G$ a discrete group) where cofibrations are $G$-CW-inclusions and weak equivalences are simple homotopy equivalences, i.e. equivariant homotopy equivalences with trivial Whitehead torsion in $Wh(G)=K_1(G)/\{\pm g\\;;\\;g\in G\}$. This class of weak equivalences is not strongly saturated, therefore it does not come from a model category. It would be very interesting to check whether this is a Brown category (probably easy) and whether some lifting axiom fails.

Michael Weiss considers in his paper "Hammock localization in Waldhausen categories" the Waldhausen category of $G$-CW-spectra ($G$ a discrete group) where cofibrations are $G$-CW-inclusions and weak equivalences are simple homotopy equivalences, i.e. equivariant homotopy equivalences with trivial Whitehead torsion in $Wh(G)=K_1(G)/\{\pm g \mathrel{;} g\in G\}$. This class of weak equivalences is not strongly saturated, therefore it does not come from a model category. It would be very interesting to check whether this is a Brown category (probably easy) and whether some lifting axiom fails.