Steps to reproduce:

There are two users User A and User B.
User A has Administrator rights and has the read-write permissions to Group A.
Group A includes Host A.
The Trigger Actions has a rule for sending a message to User B in case of problems on the Host A.
User A cannot see this rule.

In order for User A to see this rule, I need to create Group B and include there User A and User B, nothing more is required.
This behavior seems strange, and I can't find a section in the documentation that fully covers the details of user group membership and host group permissions for managing Action triggers.

It turns out that an Admin who has rights to the Host cannot fully manage mailings.

Looks like a bug, but I'm not sure. Can you review the request and tell if this behavior is expected?