It is well known that a sufficient and necessary condition for a continuous function g g to be almost periodic on time scale T {\mathbb{T}} is the existence of an almost periodic function f f on R {\mathbb{R}} such that f f is an extension of g g . The purpose of this article is to extend these results to S p {S}^{p} -almost periodic functions. We prove that the necessity is true, that is, an S p {S}^{p} -almost periodic function on T {\mathbb{T}} can be extended to an S p {S}^{p} -almost periodic function on R {\mathbb{R}} . However, a counterexample is given to show that the sufficiency is not true in general. By introducing a concept of minor translation set and characterizing the almost periodicity on T {\mathbb{T}} in terms of this new concept, we obtain a condition to ensure the sufficiency. Moreover, we show the necessity of this condition by a counterexample.
