Short answer: Only partly. The plan can allocate duties to a TPA or another named fiduciary, but the employer remains responsible for prudently choosing that vendor and monitoring it, and keeps any duties it did not hand off.
ERISA lets the plan document set up a procedure for allocating fiduciary responsibilities and designating others to carry them out. When that is done properly, the named fiduciary is generally not liable for the other person’s acts, except where it was imprudent in making the allocation or designation, in setting up the procedure, or in continuing it.
That last point is why selection and monitoring never go away. An employer that hires a TPA, a PBM or a contracted Section 3(16) plan administrator should keep evidence of how it chose the vendor, what the contract delegates, and how it reviews performance and fees over time.
A practical step: build a responsibility sheet naming the plan administrator, who decides claims and appeals, who files each report and who handles stop-loss submissions, and match it to the signed agreements.