Issue
When I send a message to my email alias or to a Group I belong to, the message isn't sent to my inbox.
Example: John's primary email address is [email protected]. He also has the email alias [email protected]. If John includes [email protected] as a recipient of a message, the message isn't sent to his inbox. However, if John sends a message to his primary account, [email protected], the message is sent to his inbox.
Resolution
To prevent clutter, Gmail doesn't deliver messages that you send to your own alias (or to a Group you belong to) to your inbox. You can find the message in Sent Mail or All Mail. To get these messages in your inbox, set up the alias or Group as an alternate "Send mail as" address for your account.