'Edit' and 'Delete' Actions in Tasks Not Found in Lightning Experience

If you have confirmed that the impacted user has the necessary Edit and Delete permissions, either via a profile or permission set, check to ensure their page layout includes the expected buttons in the Salesforce Mobile and Lightning Experience Actions. 
 

1. Navigate to Setup within Lightning.
2. Click the Object Manager tab.
3. Go to Task > Page Layouts > click on the respective Page Layout Name link.
4. Click Mobile and Lightning Actions.
6. Add the Edit and Delete buttons to Salesforce Mobile and Lightning Experience Actions section.
7. Click Save.

Note: If the 'Salesforce Mobile and Lightning Experience Actions' section has this note: "Actions in this section are predefined by Salesforce...", click the override the predefined actions.
 

See Also
Customize Task and Event page layouts