With the rapid development of Internet, Internetware has attracted a lot of attention as a new promising computing paradigm. Internetware can evolve continuously during runtime with the changes of its environment, thus to achieve higher software performance and customer satisfaction. So it is important to develop effective Internetware context awareness technologies. In this paper, we propose a novel multilayered context awareness technology. First, we build a three-layered architecture for Internetware to avoid disorderly context information transmission. Then, we utilize the ontology model language OWL to formalize Internetware context, to form a unified understanding of context information in the whole Internetware system. Finally, an extensible context management framework is proposed to drive different components in the Internetware system to access and operate related context information. In this proposed context management framework, we utilize the publish/subscribe mechanism to distribute context information and employ ontology libraries for context persistence. With these mechanisms, context awareness technology can facilitate Internetware evolution effectively.