CPS groups children into levels of care, according to how much "extra" they need. They've changed it around some since I was involved with them, but for ex. let's say a foster/adoptive parent or group home that takes more or less regular kids without any big challenges is level 1-2. A child who had more significant problems would need to be placed in a group home or with a foster/adoptive parent who was licensed for a higher level of care than the usual basic deal.
Again, I don't remember all the details now and it's been changed around some but as a vague example, if this teenager was in a level 3-4 facility, he would have physical, emotional, or mental difficulties that would need people with that extra training and less kids per adult. CPS could not let a regular foster parent take that child in, they'd have to get the advanced certification first. The state also pays more per month per child for the ones with more intensive needs.
I forget what the highest numbered level of care is but that would be a child who had very serious extra needs, like say needed to be on a machine to be able to breathe, or had had so many suicide attempts that he/she had to be in a facility where she was watched every minute of the day or night.
But even a level 1-2 teenager in care who has had a rough life, can definitely be quite a handful and have assorted problems beyond what most of us would probably consider a regular, well adjusted teenager.
So, say your kid was in a regular group home with other teenagers. He could be there and have had hospitalizations for mental or behavioral problems, a stay at juvenile hall, and so on. (Oh that's another thing, kids can and do get their levels of care changed. For ex., if your kid did well in his level 3-4 facility for six months or so, it wouldn't be unusual at all that he'd then qualify for placement in a level 1-2 group home or regular foster home placement).
Maybe the woman who wants to take him in is a friend of his family and decides she wants to help, or she could be a teacher, minister, or nurse who got to know him through her job, or just someone who for some reason decides she wants to take in him in particular, or one or more foster teens in general, rather than this particular child.
Her reasons might add more interest to your story. For example, maybe she has this interest because she had been a foster child herself at one time. I was in foster/adoptive certification classes with a couple who were there because they wanted to take in their teenage daughter's friend who was in the system. Another couple wanted to take in three kids who were somehow related to them but if they got certified, they'd qualify for the money to help raise the kids, which at the time would have been like $1,500 a month plus their medical insurance and other benefits. If they had just taken them in without getting certified, they wouldn't have the state into their business nearly as much but they also wouldn't get the extra financial help. I believe the only difference with a single foster parent as opposed to a couple is that they may not be allowed to have as many children as a home with two parents. It is (or used to be) six kids total for a couple, including your own children. And, I believe the number was four total for a single parent. Sometimes a single parent home is preferred, say for a child who has been abused by an adult of the other gender and fears them or whatever.
Maybe you could just go ahead and write your story, and then have some of us with some familiarity with the system look it over and tell you our ideas of how to correct anything you might have put in it that wouldn't be likely to really happen.
Also, imo sometimes the best way to get around a lot of the technical details is to just not mention many of them. I think it would be pretty easy for you to swing having this happen without explaining much of how it happened and the readers would buy it. The technical details aren't all that interesting anyway, imo. Good luck with it.