Ok, originally my problem was rooted—I thought—in moving a site to a new hosting service. I tried various things. But before I posted this, I decided I would try it with a brand-new fresh install, and I get the same error message. So it can't be due to my old files which is what I thought was the problem.
When I try to install K2 I get
JInstaller: :Install: File does not exist /home/mxxx/public_html/xxx/tmp/install_5d59d98726f2e/components/com_k2/k2.php

[This was what I was first going to post.]
Originally, I moved all the folders and files and imported the database I exported from the old site.

There is an error message in the display area where K2 would appear on the main administrator page (An error has occurred.
0 Class 'K2HelperHTML' not found). I tried reinstalling K2, uninstalling, reinstalling. I tried fresh files and folders of Joomla and installing K2.
When I try to install I get
JInstaller: :Install: File does not exist /home/xx/public_html/xxx/tmp/install_5d59d98726f2e/components/com_k2/k2.php
Component Install: Failed to copy site files.

here is the system information:
PHP Built On Linux alabama.shnw.net 3.10.0-962.3.2.lve1.5.25.10.el7.x86_64 #1 SMP Wed May 29 04:37:40 EDT 2019 x86_64
Database Type mysql
Database Version 5.5.5-10.1.41-MariaDB-cll-lve
Database Collation utf8_general_ci
Database Connection Collation utf8mb4_general_ci
PHP Version 7.2.21
Web Server Apache
WebServer to PHP Interface litespeed
Joomla! Version Joomla! 3.9.11 Stable [ Amani ] 13-August-2019 15:00 GMT
Joomla! Platform Version Joomla Platform 13.1.0 Stable [ Curiosity ] 24-Apr-2013 00:00 GMT
User Agent Mozilla/5.0 (Macintosh; Intel Mac OS X 10_13_6) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/76.0.3809.100 Safari/537.36

I also did a fresh install using Joomla 3.9.8 and get the same error message when trying to install K2.

I increased the upload limit from 2 to 4 for something else. But I did try to install it from the web first using "install from web" and then I tried to upload the package with the same result.

What can I do to find out if the permissions issue? The tmp folder is 755.

Thanks you.
Patti

I contacted the hosting support, and they tried to install K2, and ended up doing it by uploading a package, but still got error messages. They are asking me to ask you what server side adjustments need to be made. here is their message:

I am was checking the issue for the last few hours. We could see that the K2 installation shows successful when we do via the admin > Extension > Manage > Install > Upload package

We used K2_v2.9.0.

But still, the issue you have mentioned is present under the website.

Following is the error we get on the log file.

[23-Aug-2019 14:50:46 UTC] PHP Notice: Use of undefined constant K2_JVERSION - assumed 'K2_JVERSION' in /home/missouri/public_html/16juliadentgrant/administrator/modules/mod_k2_stats/mod_k2_stats.php on line 15
[23-Aug-2019 14:50:46 UTC] PHP Notice: Use of undefined constant K2_JVERSION - assumed 'K2_JVERSION' in /home/missouri/public_html/16juliadentgrant/administrator/modules/mod_k2_stats/mod_k2_stats.php on line 23
[23-Aug-2019 14:50:46 UTC] PHP Notice: Use of undefined constant K2_JVERSION - assumed 'K2_JVERSION' in /home/missouri/public_html/16juliadentgrant/administrator/modules/mod_k2_stats/helper.php on line 23
[23-Aug-2019 14:50:46 UTC] PHP Notice: Use of undefined constant K2_JVERSION - assumed 'K2_JVERSION' in /home/missouri/public_html/16juliadentgrant/administrator/modules/mod_k2_stats/tmpl/default.php on line 15
[23-Aug-2019 14:50:46 UTC] PHP Notice: Use of undefined constant K2_JVERSION - assumed 'K2_JVERSION' in /home/missouri/public_html/16juliadentgrant/administrator/modules/mod_k2_stats/tmpl/default.php on line 32

Also, the Class 'JPane' not found issue indicates to the templates issue. Is it possible to have a check with your developer and let us know if any server-side adjustment is needed?

The JPane error he refers to is found in the administrator landing page where K2 components are usually listed. See screenshot attached. Thank you.

The solution "wasn't as simple as file permissions." The hosting company was able to determine that the problem lay with the "handler." The server using the FCGI handler would not install K2, but the server using ISAPI did. They moved my site to to the server with the ISAPI handler, and now all is right with the world.