Cloud computing provides cost effective solutions for deploying services and applications. Even though resources can be provisioned on demand, they need to adapt quickly and in a seamless way to the workload intensity and characteristics and satisfy at the same time the desired performance levels and the corresponding SLAs. Autoscaling policies are devised for these purposes. In this paper, we apply a state-of-the-art reactive autoscaling policy to assess the effects of deploying the HTTP/2 server push mechanism in cloud environments. Our simulation experiments — that rely on extensions of the CloudSim toolkit — have shown that the autoscaling mechanism is beneficial for web servers even though pushing a large number of objects might lead to server overload.