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Record page settings are currently set to related lists.
This means that users do not have access to the activities component. However, they do have access to the chatter publisher component. Is there any way to display a custom new task quick action in the chatter publisher?
P.S. - The user interface is Lightning.
if you are looking for a solution where both related list for activities and activities component should work in parallel, yes it is possible.
Salesforce has given you a standard component called Related List Single. All you need is to add related list single component to Lightning page layout using Lightning App Builder and select Open Activities in related list names available on the right-hand side panel. Once done, Save the page and activate it for the users you want.
This will let you see the Open Activities Related list with actions to create Task and Event with record type selection functionality. In the same way, You can repeat the process for Activity History related list.
This functionality gives you the flexibility to use both activities component and related list for task and event.
Read my blog on more information:- OPEN ACTIVITIES AND ACTIVITY HISTORY RELATED LIST IN LIGHTNING EXPERIENCE
