Consider homomorphisms ϕ : X → Y and ψ : Y → Z, where ϕ is open and N-to-one, ψ is almost periodic. In the paper by R. J Sacker and G. R. Sell, it was shown that, under a certain condition on the phase group, the composition ψ ∘ϕ : X → Z is almost periodic (provided that Z is trivial and X is minimal). In this paper almost periodicity of ψ ∘ ϕ is studied under connectedness conditions on the fibers of ψ. For instance it is shown that if ψ is almost periodic with connected fibers than ψ ⋅ ϕ is almost periodic. If ψ is locally almost periodic with locally connected fibers then ϕ ∘ ψ is locally almost periodic.