Thanks for your answer.
There is no API guide for executing Butler card button on [REST API guide page](https://developer.atlassian.com/cloud/trello/rest/api-group-actions/).
But I belive there is a way to execute Butler card button as you said.
So I am analyzing Network tab on DevTools with chrome.
If I resolve this problem I will share for you.
I've tried to request with below data.