Git: is there a way to auto push a subtree?

I have a big private repository which is maintained on a local network. I'd like to automatically push a subtree of that repository outside of that network. I need it to be simple:

*Task* someone pushes to local remote repository --> a subtree is automatically pushed to some other repository

I am not sure if this could be achieved with a server side hook because AFAIK there is no such thing as pushing subtrees from bare remotes. I came up with two ideas:

I could clone the remote on the server and automatically split the subtree in the cloned remote. This doesn't really help because I don't know how to auto-pull the subtree (others also have this problem).
Another idea is to write a custom client-side post-commit-hook and make every user install it, but this is terrible, isn't it? Git book specifically states that policies should be enforced on server side.

Is there a simple way of achieving something like this? Or is this impossible and it's just git abuse?

Solution

Umm, I'm a bit embarrassed. Apparently this was much easier than I thought. Here is a hasty solution which builds on @wrzasa suggestion:

Clone your repository on the server to which you are pushing, like this (dir.git is a bare repo):
```
.
|- dir.git
|- dir
```
In dir do: git remote add <remote-name> <remote-address>

In dir.git/hooks/post-receive put:

#! /bin/bash
unset GIT_DIR
cd ../dir
git pull ../dir.git
git subtree split --prefix=<subdir-in-dir> --branch=<branch-name>
git push <remote-name> <branch-name>

Remember to make post-receive executable. See this answer if you wanna know why unset GIT_DIR is needed.

That's pretty much it. Now whenever someone pushes to dir remote (i.e. dir.git) subtree under <subdir-in-dir> will be pushed to <remote-name>.