Symptoms

Although these users are a part of the Sharepoint group with 'Solution Designers' permission and the group sync was performed via 'K2 for Sharepoint Settings > Synchronize Groups', these users still experience the error: Access Denied To design K2 applications, you must be a member of a Sharepoint group that has K2 Solution Designer permissions. If you have recently changed Sharepoint group membership, please contact your K2 Administrator to use the 'Synchronize Groups' option. This does seem to work in other site collections.
 

Diagnoses

It was determined that the site title in question made use of the colon (:) character. The site title is used when provisioning K2 Group Providers and was failing to resolve these Sharepoint group due to the colon characters. If the site was named 'StringA: StringB', the following error would be logged: 64010 The Identity Group:SP: StringBPortal Owners, does not appear to be discoverable or does not exist. 64007 Provider did not return a result for SP: StringBPortal Owners on GetGroup It appears the 'StringA:' portion was truncated.
 

Resolution

The site collection was deleted and recreated without the colon character. This allowed for successful deployment, activation, and syncing of the Sharepoint group for access to the 'K2 Application' without any Solution Designer permission issue. There are some special characters that should avoided when creating site, lists, libraries as per: https://support.microsoft.com/en-us/help/905231/information-about-the-characters-that-you-cannot-use-in-site-names,-folder-names,-and-file-names-in-sharepoint