Hey @Herve_Buisset
Thanks for clarification.
Indeed, with the Gmail addon, there is no button to add the complete email to an existing task.
But you could technically copy/paste it in the task’s comment, directly from Gmail (but still a bit “dirty”).
With Outlook, this is apparently possible (as stated on the integration page)
See text
Add emails to existing task
Your entire team probably isn’t copied on every email chain that includes information they need to do their jobs. But without these details and context, they can’t effectively do their work. Now you can add the email to an existing Asana task as a comment right from Outlook by clicking Related Tasks in the Outlook top bar, searching for the task you’d like to add the email to, and then click Attach Message.
Or you could use a nice 3rd party tool like Supersana - cc @Bryan_TeamKickstart