K2 v2.2 was released around 2010, so it's quite old. Here's the process to follow:

- First grab a physical backup of the site and database. Just in case.
- The upgrade K2 from v2.2 to v2.6.9 using this zip file: getk2.org/downloads/?f=K2_v2.6.9.zip
- After that, install K2 v2.10.2 from: getk2.org/downloads/?f=K2_v2.10.2.zip

K2 v2.10.2 is fully compatible with Joomla 2.5.28 (latest).

After that, you have 2 options:
- If you want to start fresh with Joomla 3.x but keep your K2 data (from the Joomla 2.5 site), you can install a new separate site with Joomla 3.x and K2 v2.10.2, then (using phpMyAdmin) delete all K2 tables from the NEW site, export the K2 tables ONLY from the Joomla 2.5 site and finally import them into the new Joomla 3.x site. Finally, copy the static assets (images, videos, attachments etc.) of the Joomla 2.5 site from /media/k2/ to the new Joomla 3.x site's /media/k2.
- Upgrade Joomla 2.5 to 3.x using the official guide: docs.joomla.org/Joomla_2.5_to_3.x_Step_by_Step_Migration - this will allow you to keep your menus, modules etc. into Joomla 3.x. Since K2's database tables are identical in all Joomla versions, K2 will work out of the box.

As a sidenote, you will most definitely have to update your K2 template overrides (if you use any) and re-save your category, menu and module settings that relate to K2.

2.6.9 will perform some changes to the K2 database tables. Then you can update to 2.10.2.