Git submodule on remote bare

I've setup my environment so I can push to a remote bare repository. I used these commands to set up the remote repository:

$ mkdir ~/website.git && cd ~/website.git
$ git init --bare

And

$ cat > hooks/post-receive

#!/bin/sh
GIT_WORK_TREE=/var/www/website git checkout -f

$ chmod +x hooks/post-receive

And on my local environment:

$ git remote add web ssh://website.com/home/website.git
$ git push web +master:refs/heads/master

Now I can deploy to this remote using git push web, and everything works great..

The problem: Submodules

I have a few submodules on my project that aren't getting initialized/updated at the remote repository. I can't run git submodule update on the bare because it's bare, and I can't run it on the /var/www/website folder because it's just a copy of the files and not a git repo.

One possible way might be:

• to setup /var/www/website as a (non-bare) repo
• have your post-receive hook of your bare repo:
  • set GIT_DIR and GIT_WORK_TREE to the non-bare repo at /var/www/website
  • cd /var/ww/website
  • git pull ~/website
  • git submodule update (a bit like in "How do I init/update a git submodule in a working tree after pushing to a bare working directory?")

In other words:
Pull from the bare repo instead of trying to checkout from a bare repo: a non-bare repo should be able then to accommodate the git submodule update step.

An example script may look like

#!/bin/sh

# Get the latest code
cd /path/to/bare/repo

# Set git variables
GIT_WORK_TREE=/var/www/website
GIT_DIR=/var/www/website/.git

# Go to website and pull
cd /var/www/website
git pull /path/to/bare/repo
git submodule update --init --recursive

# Run additional build stuff here