There were two things related to the problem, try **Solution-1** and if you still don't see the actions then **Solution-2** should do it (as was the case with me).

### [1 - Add "New Task/Event/Email" buttons to the object Page layouts](https://help.salesforce.com/articleView?id=activitytimeline_configure_call_task_event_tabs.htm&type=5)

 - View the page layout for the object you want to update. Make sure
   buttons for New Task, New Event, Log a Call, and Send an Email (not
   Send Email) are added to the Salesforce Mobile and Lightning
   Experience Actions section of the page layout.
 - Make sure your users have permission to create new tasks and events,
   log calls, and send email.

### [2- Edit Change global actions](https://salesforce.stackexchange.com/questions/181814/new-task-action-missing-in-lightning-experience)

 - In the Lightning Experience, Setup Home | User Interface | Global
   Actions | Global Actions
 - Click Edit next to any of the missing global actions (New Event, New
   Task, Log a Call)
 - Look at the 'Record Type' pick list and select any value but **--Master--**
  
*Note: Solution-1 didn't work because I had multiple record-types for activities.* 

Above solution is valid not only for [Lightning-for-Gmail](https://trailhead.salesforce.com/en/modules/gmail_integration/units/gmail_integration_unit_2) but also for LEX users.