PEO stands for "Professional Employer Organization", industry jargon for a particular form of HR outsourcing. At Insperity we refer to the relationship as co-employment. This is an IRS construct that permits the service provider to accept responsibility as the administrative employer of record for things such as payroll taxes, worker's comp, and benefit plan sponsorship, including 401k at times.
If the client chooses to participate in Insperity's multiple employer plan, they no longer have responsibility for administration or compliance, and there can be cost savings as well. Most clients choose this option for simplicity since the plan is integrated to payroll.
Alternatively the client may remain plan sponsor, with Insperity as recordkeeper and administrator. With this option the client may retain their financial advisor and there is greater flexibility over plan design.
If your client kept their existing 401k, they would remain as plan sponsor under their own tax ID even as the employees are paid under the tax ID of the PEO.