How to make security group in one forest show up in another forest?
Different group types have different "visibility" in multi-domain and multi-forest environments, as you've found (Microsoft can provide more detail). Global Groups, for example, are only "visible" within the domain in which they reside and can only contain users from that domain (because of how the security-identifiers of the member users are stored).
Microsoft's best-practice guidelines are as follows:
Create a global group in each domain to contain members from that domain corresponding to a job role
Create a domain local group in the domain with the resource to be controlled and grant the domain local group permissions to the resource
Nest the global groups from each domain into the domain local group
In some scenarios a universal group can come into play, too (when the resources to be managed are distributed throughout multiple domains, typically).
There some nice (albeit, w/ a funny aspect ratio) pictures in this Microsoft TechNet forum thread that can give you some background. I'd also advise looking at the Wikipedia article form some more background, as well.