One more point for information:
When I start the template update on a J4 website, I get this error:
Unable to extract file: download.php_element_tpl_t4_bs5_blank_version_1.1.3_type_template_coreVersion_j31
So it looks like the J4 site is getting an update for a J3 template.
After that I have Framework & Free Templates & ExtensionsT4 Framework
T4 Blank Template Bs5 v. 1.1.3 downloaded again and installed manually. On the J4 website I have now been able to perform the update without the error.
Things are not going well on J3 websites as indicated at the beginning of this support ticket.
By the way, I think Frank made a mistake in the title of this topic.
The T4 system plugin has version number 2.1.0 and the T4 Blank Templayte BS5 has version number 1.1.3