If someone's kids are in foster care can the mother relinquish her rights to someone before cps takes her rights

Typically, if they are in foster care they will stay with the current foster parents if there is no approved family to take over care.

The "friend" would have to undergo a rigorous background check.

You do realize that terminating your rights, either willfully or forced by the court, is the first step in paving the way for your children to be adopted, right? That would mean you no longer have any rights to the children at all. None. You will only be bonded with them by blood, not by law.

If your friend is willing to foster or adopt your children, they have to apply to CPS. If the parents are willing that will help.

So much depends on your friend. If they think the application is part of an attempt to help you be with them when the court ordered them taken (neglect, crime, drugs) then they won't approve it.
You don't have to 'relinquish your rights' in most cases. That's a separate issue.

Isn't there a father somewhere? What about his rights?