Hello Erick,
For essential variables of pre-qualified procedures in AWS D1.1 : 2015, use table 3.7.
There you will see - "3) A change in base metal group number(s) (see Table 3.1) " as an essential variable.
So , the answer to your question is yes, the same PWPS may be used for base materials in the same Group.
However, you may not want to.
The difference between A572 and A588 is that A588 is a weathering steel with special alloying elements to provide a corrosion limiting patina, typically used where the weldment is desired to be left unpainted.
When this material is used, often a particular filler metal is used for patina color matching properties.
If this outcome is desired, a separate PWPS would also need to be used that lists the desired filler metal.
See this link for more info -
https://www.hobartbrothers.com/news/181/523/Welding-Weathering-Steels-Understanding-the-Material-and-Choosing-Filler-Metals-for-the-Job.htmlAnd again, a separate Welder Performance Qualification (WPQ) would not be required, even if you made an ASTM 588 specific PWPS, unless there are changes in the joint types, positions or base metal thicknesses not covered under the original WPQ.
Tim