You can't do this with only group system. You should use a bot and its deep linking ability provided by Telegram Bot API. When your website redirects your user to your bot, it should have this deep-link. Give a token to this deep-link, generated for the user. So your bot can understand who is this user. When user touches to the 'Start' button of your bot to start the conversation, your bot will identify the user. Then show a message to the user, say "Please join this channel to complete the task. {group link}".
Add your bot to your group too. And watch new users came your group. If the user who came with deep linking to your bot and the new user who came to your group matches, then it's what you want.
I hope that will give you an idea. The main solve is that I guess, as far as I know with 2+ Telegram bot development experience.