By default, when Intercom sends email notifications to your users or leads, it includes a summary of the conversation so far, and the recipient can reply directly to continue the conversation. To exclude this conversation content from notifications and keep the conversation in the Messenger, go to your security settings page, and uncheck the box for "Allow conversations to continue over email":
With this setting disabled:
If the conversation was started in the Messenger on web, the email notification will link to the webpage with the messenger installed.
If the recipient has never visited a webpage (on your site, or in your web app) with the Messenger installed, no email notification is sent. For example, if a user has only interacted with you in your mobile app.
In these cases, we recommend that you enable Intercom push notifications in your app.
Group conversations are not supported with content in notifications disabled, so your customers cannot add people to conversations by CCing them in an email, and your team cannot add additional participants in the Inbox.