Much better for you to bring the cat back to your house and try to adopt it out yourself, if they plan to euthanize it at some point. Sounds like that is your plan. I couldn't set my little feral free outside because I realized after I picked her up that she was only about 4 months old and tiny. Originally I planned to socialize her and then find her a permanent home. After about 10 days, I decided she had a permanent home with me. I seem to fail at "temporarily" helping cats.