Homogeneity problems in the theory of Čech compactifications

If X is a completely regular topological space, there exists a space sX, the so-called Cech compactification of X, which is characterized by the following three properties: sX is a compact (bicompact, in the older terminology) Hausdorff space, X is a dense subset of sX, and every bounded continuous real-valued function on X can be extended to a continuous function on βX [1; 831].