Joomla 3.10.17 LTS shipped a fix that changes how Joomla creates the pagination links. The change destroys k2 pagination
3.10.18 LTS release (next tuesday) will add an option to restore the old behavior but the fixed issue (a cache poisoning attack) is re-introduced.

You should downgrade joomla version to 5.12 and wait next joomla update. I faced with this issue too

Hello!
After the Joomla 3.10.17 LTS update, the pagination in the K2 component no longer works.
The problem is in /libraries/src/Pagination/Pagination.php.
It was not solved in the new update Joomla 3.10.19 LTS
Before updating to Joomla 3.10.17 LTS the following page is displayed:
/index.php?option=com_k2&view=itemlist&layout=category&task=category&id=1&Itemid=124&limitstart=12 (Works)
After updating to Joomla 3.10.17 LTS the following page is displayed:
/index.php?option=com_k2&view=itemlist&layout=category&id=1&Itemid=124&limitstart=12 (It doesn't work). It's missing &task=category

yes 3.10.19 LTS pagination settings still not working
Global Configuration/System Settings/Legacy pagination behaviour

Joomla 3.10.19 LTS K2 pagination FIX (from Joomla LTS Support):

Set System/Global Configuration/System Settings Legacy pagination behaviour to YES
edit view html php in
components/com_k2/views/itemlist/,

old:
if ($document->getType() != 'json') {
// Pagination
jimport('');
$total = (count($items)) ? $itemlistModel->getTotal() : 0;
$pagination = new JPagination($total, $limitstart, $limit);
}

After:
if ($document->getType() != 'json') {
// Pagination
jimport('');
$total = (count($items)) ? $itemlistModel->getTotal() : 0;
$pagination = new JPagination($total, $limitstart, $limit);
$pagination->setAdditionalUrlParam('task', $task);
}

Thank you Kostas for your help. This works.
It also works without activating legacy pagination behaviour.
So could this fix please be merged?

hi, the problem to me persists. adding &task=category nothing change

The problem is in the URL . I don't know how rewrite it. try to explain better

When I click the system use the default link
www.xxx/component/k/?view=itemlist&layout=category&id=15&Itemid=1395&start=48

and I have error 404 Component not found

If I use option=com_k2, it works

www.xxx/?option=com_k2&view=itemlist&layout=category&task=category&id=15&Itemid=1395&start=48

how can I fix, please? thank you

i have the samme problem with joomla 3.10.12

After  k2 v2.11.20240911 update, pagination stop working again
edit view html php in line 594 components/com_k2/views/itemlist/
and works again 
Joomla 3.10.19 LTS (with Global Conf/ System / Legacy pagination behaviour YES)

 if ($document->getType() != 'json') {
            // Pagination
            jimport('');
            $total = (count($items)) ? $itemlistModel->getTotal() : 0;
            $pagination = new JPagination($total, $limitstart, $limit);
            $pagination->setAdditionalUrlParam('task', $task);
        }

After k2 v2.11.20241016 update pagination stop working again...
 view html php in line 594 components/com_k2/views/itemlist/

        if ($document->getType() != 'json') {
            // Pagination
            jimport('joomla.html.pagination');
            $total = (count($items)) ? $itemlistModel->getTotal() : 0;
            $pagination = new JPagination($total, $limitstart, $limit);
        }

With this change:
        if ($document->getType() != 'json') {
            // Pagination
            jimport('joomla.html.pagination');
            $total = (count($items)) ? $itemlistModel->getTotal() : 0;
            $pagination = new JPagination($total, $limitstart, $limit);
            $pagination->setAdditionalUrlParam('task', $task);
        }

Pagination works again BUT pagination on k2 TAG menus NOT working
Wrong url
site.url/tagmenu/itemlist/tag?start=10
Instead of Correct url
site.url/tagmenu?start=10
can someone help?
Joomla 3.10.19 lts

I remember you telling us how good Joomla 3 is compared to the bad 4 & 5. We have been forced to pay support for Joomla 3 LTS but after every update we are looking for solutions to get it working. on the contrary, for months now I have never encountered any problem with my other sites in joomla 5
 

solution from Joomla 3 LTS support:
components/com_k2/views/itemlist/
view html php  line 590

change from
        if ($document->getType() != 'json') {
            // Pagination
            jimport('joomla.html.pagination');
            $total = (count($items)) ? $itemlistModel->getTotal() : 0;
            $pagination = new JPagination($total, $limitstart, $limit);
      }

to 

        if ($document->getType() != 'json') {
            // Pagination
            jimport('joomla.html.pagination');
            $total = (count($items)) ? $itemlistModel->getTotal() : 0;
            $pagination = new JPagination($total, $limitstart, $limit);
            $pagination->setAdditionalUrlParam('task', $task);

            if ($task === 'tag') {
                $pagination->setAdditionalUrlParam('tag', JRequest::getString('tag'));
            }
      }