Typically, if a user sends mail from Gmail, the mail can be checked in the Inbox.
Comprehensive mail storage function allows you to save emails as senders’ sent email when Gmail automatically sends email with SMTP Relay.
If you’re using Google Vault, you can also save email sent with SMTP relay or Google Services in Google Vault. Learn more│Google Guide
Go to Google WorkspaceSetting for GmailComplianceComprehensive mail storage, and activate.