For each service that HR delivers, if there is a realistic option to outsource it then there is a case to make to assess whether this would deliver better service and/or cost savings. Any such analysis should result in a business case that has clear financial rigour but, equally, a solid basis by which the required quality of service is assured.
It is possible to outsource pretty much the whole HR Function to a Business Process Outsourcing service provider, as many Organisations have explored to varying degrees in the last 20 years. But for any component or wholesale outsourcing arrangement it has to be remembered that managing suppliers against contracts (and periodic contract renewals, disputes, etc) is a very different skillset to the traditional running of a HR Function and meeting this capability must be part of any such consideration.