In the US, typically the A/E design team will specify the garage layout, aesthetic requirements, spans, locations of columns, etc. These are usually worked out with a precaster early in the project. The A/E documents then show a foundation system based upon a presumed garage design that will be verified once the precaster provides the final shop drawings.
So in a sense, a lot of the garages done by A/E design teams are similar in nature to how pre-manufactured metal buildings are done - the EOR does the foundation and oversees the overall design - and delegates the design tasks of the PC garage above the footings to the precaster and the precast engineers either working within their firm or sub-contracted out.
For fees - it depends on if you are the overall EOR/foundation designer or the precast engineer.