Changing user agent after click

I'm wondering if there is any way to change user agent after navigating from one web page to other? It's little bit hard to explain what I want to achieve, but let's visualize this: Website1 is my website, I can modify this source code how I want. Website2 is external website, nothing could be changed there.

1. Enter website1. $_SERVER['HTTP_USER_AGENT'] shows: useragent1


2. Click on link on website1 (which points to website2)


3. Here "magic thing": useragent1 is changed to useragent2


4. Website2 loaded. $_SERVER['HTTP_USER_AGENT'] (on website2) shows: useragent2

Is this possible to accomplish "magic" point 3? How to do that?