Torus-like continua which are not self-covering spaces

For each non-quadratic p-adic integer, p > 2, we give an example of a torus-like continuum Y (i.e., inverse limit of an inverse sequence, where each term is the 2-torus T² and each bonding map is a surjective homomorphism), which admits three 4-sheeted covering maps f₀: X₀ → Y, f₁: X₁ → Y, f₂: X₂ → Y such that the total spaces and X₀ = Y, X₂ are pair-wise non-homeomorphic. Furthermore, Y admits a 2p-sheeted covering map f₃: X₃ → Y such that X₃ and Y are non-homeomorphic. In particular, Y is not a self-covering space. This example shows that the class of self-covering spaces is not closed under the operation of forming inverse limits with open surjective bonding maps.