Lorentz Transformation and time dilatation

We consider two inertial frames S and and suppose that frame moves, for simplicity, in a single direction: the X -direction of frame S with a constant velocity v as measured in frame S. Using homogeneity of space and time we derive a modified Lorentz Transformation (LT) between two inertial reference frames without using the second postulate of Einstein, i.e., we do not assume the invariant speed of light (in vacuum) under LT. Roughly speaking we suppose: (H) Any clock which is at rest in its frame measures a small increment of time by some factor s=s(v). As a corollary of relativity theory (H) holds with Lorentz factor 1/γ. For s=1 we get the Galilean transformation of Newtonian physics, which assumes an absolute space and time. We also consider the relation between absolute space and Special Relativity Theory, thereafter STR. It seems here that we need a physical explanation for (H). We introduce Postulate 3. The two-way speed of light in and -directions are c and outline derivation of (LT) in this setting. Note that Postulate 3 is a weaker assumption than Einstein's second postulate.