S.K.
In Missouri and California, group home just means that they are licensed for a larger group and have help. It's a bit of a half step between the small, private in-home daycare and the big and sometimes impersonal center. Group homes in Missouri were especially designed for the small cities and towns that don't have a lot of options for daycare. But anyone can apply if they and their home can meet the guidelines.
I have been a family caregiver for over 20 years. I have taken children in after being in a center and hating it. I've also had children that just were not happy enough in family daycare end needed the large and rowdy environment of a center. Only you know your child best. If your child is outgoing, tends to make friends easily, doesn't have a lot of special needs that could fall through the cracks in those places, likes to have quite a few new experiences, enjoys the table activities and preschool work they would be doing in a more structures way, then a daycare center or group home may be good.
If your child is slow to warm up, needs lots of hugs and some lap time, has special fears or quirky behaviors, a family daycare would be the better bet.