I want to add a task (action / behavior) to a specific category.

In more detail:

The category store a list of URLs.
I want to add the ability to do something with this list.
so in the list I added a "button" that will activate the task.

The link will have a format of:
my-site.com/k2-demo/index.php?option=com_k2&&layout=category&task=myNewAction&id=2&Itemid=488

What is the procedure to do it?

Hi. The procedure is to hire a developer to implement this for you. Or look for an existing extension that may do what you need.

Lefteris Kavadas wrote: Hi. The procedure is to hire a developer to implement this for you. Or look for an existing extension that may do what you need.

I am a programmer...
And believe me that I check the existing extensions and didn't find ...
I also try to look for a tutorial about the issue of writing an extension / extra field and didn't find something that could help me. It might be that my search was not effective - so I expected to get here a pointer to such place...

This has to do with the extension and the way it is structured. Since you want add an extra task you need to create a component. And since you are a programmer you can get started at docs.joomla.org/J3.2:Developing_a_MVC_Component/Developing_a_Basic_Component.

Again, I didn't explain myself correctly... because your answer is "don't use K2", and I hope you don't mean it...

my be the word 'task' was selected incorrectly.
I will explain the exact scenario I want to implement:
1. I have two K2 categories - cat1 and cat2, each of them store different information
2. BUT the information stored in 'cat1' (let assume it is an RSS entry) is used to create entries in 'cat2'
3. I want to add this function by putting a 'button' / link in list of 'cat1' and to implement the action of "import" and store the information in 'cat2' entry

I am sure there is a way to implement it in K2...
BTW, I already implemented such a use-case in fabriks...

The answer is still the same. You can extend K2 using K2 plugins. K2 plugins are used to extend K2 item form and display the extra data in front-end. You can take a look at an example K2 plugin at github.com/joomlaworks/example-k2-plugin. However your task does require a component. Otherwise you will have to hack K2 core files, which is not a good practice since you will lose all your changes in future updates. I am not saying to use a different component. This component will simply add the task to import the data.