How to share a buildx instance with the host using Docker in Docker (DinD)

[naomichi-y]

I'm trying to share a Docker builder instance that was created on the host with a container.
However, I'm unable to see the instance created on the host from within the container. Is there a way to share these builder instances?

$ docker buildx create --name foo
foo

$ docker buildx ls
NAME/NODE DRIVER/ENDPOINT             STATUS   BUILDKIT PLATFORMS
foo       docker-container
  foo0    unix:///var/run/docker.sock inactive
default * docker
  default default                     running  20.10.23 linux/amd64, linux/386

$ docker run --rm --entrypoint docker -v /var/run/docker.sock:/var/run/docker.sock docker buildx ls
NAME/NODE DRIVER/ENDPOINT STATUS  BUILDKIT       PLATFORMS
default * docker
  default default         running v0.8.2+eeb7b65 linux/amd64, linux/386

Alternatively, if you have any suggestions for creating a persistent builder instance after running docker run, so that I don't need to create a new instance each time I run the command, I would appreciate your advice.