How to delegation windows authtication to trust domain

Question

I'm working on a feature of integration with third party system in SharePoint . We need get data from the third party API pass by windows authentication .

In SharePoint 2019 we start up c2wts and configuration delegation (same domain ) and it worked like we expect .

But it's not worked for trust domain , Let's said :
1 . Customer has two trust domain 'alpha ' and 'beta '
2 . The SharePoint host at server "alpha\sp" ,
3. We config c2wts account is "beta\user1"
Note: if the c2wts user not under domain beta will get exception " access is denied
4 . The Application pool account is alpha\user1
5 . Third party system host at server "alpha\server1"

This is our test case :
1 . Login SharePoint use domain beta account for example beta\user2
2 . Click send request button in our test solution
3 . Currently we can get the windows Identity by using SharePoint method and Impersonate the user , we confirm the user info is correct .
4 . When we use this user send request to third party system we got 401 http exceptions .

If we change c2wts account to alpha domain account and login alpha user to SharePoint everything is worked.

So guys any idea to fix the issue ? I guess maybe miss something for trust domain delegation ? please help me out : (

Thanks a lot !!!