Extensions of Mikhailova's theorem and its converse

Due to a theorem of Mikhailova the quotient space of a real vector space $V$ by a pseudoreflection group, i. e. a finite group generated by linear transformations with codimension two fixed point subspace is homeomorphic to $V$. We explain why the converse is not true in the topological category, give a complete classification of groups satisfying this property and discuss the same question in other categories.