Compact-covering-properties of finite-to-one mappings

Let X and Y be separable and metrizable spaces. Suppose that the continuous and onto mapping ƒ : X ↦ Y is countable-compact-covering. Under the assumption that, for some integer n, all the fibers ƒ−1(y) have at most n points we prove that ƒ is inductively perfect. We prove also that the finiteness of these fibers does not suffice to ensure the same conclusion.