HTTP/2 video streaming has gotten a lot of attention in the development of multimedia technologies over the last few years. In HTTP/2, the server push mechanism allows the server to deliver more video segments to the client within a single request in order to deal with the requests explosion problem. As a result, recent research efforts have been focusing on utilizing such a feature to enhance the streaming experience while reducing the request-related overhead. However, current works only optimize the performance of a single client, without necessary concerns of possible influences on other clients in the same network. When multiple streaming clients compete for a shared bandwidth in HTTP/1.1, they are likely to suffer from unfairness, which is defined as the inequality in their bitrate selections. For HTTP/1.1, existing works have proven that the network-assisted solutions are effective in solving the unfairness problem. However, the feasibility of utilizing such an approach for the HTTP/2 server push has not been investigated. Therefore, in this paper, a novel proxy-based framework is proposed to overcome the unfairness problem in adaptive streaming over HTTP/2 with the server push. Experimental results confirm the outperformance of the proposed framework in ensuring the fairness, assisting the clients with avoiding rebuffering events and lowering bitrate degradation amplitude, while maintaining the mechanism of the server push feature.