Primitive recursive plus Ackermann

Question

Let us consider the class $\cal F$ of functions that contains

• all constant functions
• all projections
• the successor function
• the Ackermann function

as basic functions, and that is closed under composition and primitive recursion.

(If we remove the line with the Ackermann function from this definition, then this is the standard definition of the class of primitive recursive functions.)

My question: What function class $\cal F$ do we get by adding the Ackermann function as basic function? Do we get all $\mu$-recursive functions, or do we get some strange sub-class?