I have worked with boys at this age who require residential placement. Some girls, too - but more as a preventative to placement. So I have a few questions:
Would you be a kinship foster home or just custody? B/c foster care would bring a good deal of services including monthly casework, law gaurdian, stipen (room/board) and medical insurance (medicaid). There would be service in place to actively reach the goal of "return to parent". Taking custody would give you more independence but little to no services and no finacial support and murky health isurance/coverage. There would be no one assigned or interested in resolving the issues that cuased your sister to need this intervention and getting her back on her feet and reunited with a bio parent.
There is no way of knowing which is "better" for a troubled 15 year old. Some group homes are wonderful, some are terrible. Does your sister WANT to live with you? Would she respect your structure and consequences? Would she continue to go to school, do chores, follow safe rules? What would happen if she didn't?
From what I've seen, you could be an amazing advocate and resource while she is in a group home. The BEST outcomes happended for children with a solid support system outside of the group home.
Being in a structured environment with the ability to earn freedom like weekend visits and holiday visits - like to a sister's house. You can be someone to show up and visit and engage in therapeutic services; someone to call and make sure she got good education and recreation - THAT is what she needs. If you can provide really solid structure, follow through and maintain consequences and rewards - maybe you could take her in. But if any of those things are better matched in a group home, you can still be a really important resource, a really important person in her life to get her on a better path.
This idea of your mom's to just "give up" is ridiculous. I've seen it ALL the time. But those children suffer terribly and have terrible outcomes. Residential services cna only do so much. !5 year olds need people to get involved and STAY really, really involved.