Hello,
if the internet works for a short time after the VPN connection establishment, than it stops, the client gets disconnected and it eventually reconnects, it seems that the VPN connection was terminated from the gateway side. The gateway would behave like this for example in case that the same license is used by more clients simultaneously.
The recent product versions (3.4.5, distributed from the Avast's website or Avast Mac Security) should be capable to handle the situation properly and to display appropriate error message.
The version 2.2.7 distributed using the App Store would behave the way you have described. The license can be transferred from one version to another using an activation code or using Avast account, both can be found in the product preferences.
In order to investigate the issue more, I would need a support package that can be created from the application's help menu. It contains private data like a license, so we can discuss it in a private message.
Best Regards,
Martin