Role based permissions and what items they affect are not contained on thr role itself.
All security is actually an attribute/field on the item itself, and only maps to a role.
The only data that a role has is to map which users are members of a specific role.
You will need to package up the Content Items, in addition to the role (if the role isn't already installed), and install those items on the target site.
Use of roles should be confined to only the few items that you need, and allow the rest to occur through inheritence.