While button automations don’t currently support that exact combination in 1 automation, you could accomplish this but it would require multiple automations on both boards.
Off the top of my head, a combination of (1)button click to create an item in the target board that would then (2)trigger the creation of a group. From there you can have (3)another automation that duplicates or creates a group. But depending on where that group info is coming from and your specific workflow, this could get complicated.
But an automation like this would be easily achievable through Integromat or Zapier and much more customizable. If you do decide to go that route please see this thread on how to send a Webhook via a Button Column click. If you need further assistance setting this up, I use Integromat and could help troubleshoot your scenario.
Hope this helps!