Why can't a user send an email from a public mailbox (MS Exchange 2010)?

Question

System administration

Hello. A newly created user in the domain (haha.ru\vasya) should be able to send mailouts on behalf of the company's "[email protected]" public mailbox. How it looks like: on the Exchange 2010 server, the command "Add-MailboxPermission -Identity [email protected] -User vasya -AccessRights Fullaccess -InheritanceType all" is executed, and in the outlook (2007, 2010) configuration on the client PC, the default mailbox is registered - mail @haha.ru. The user vasya enters the domain, opens the mail client without any problems, sees the only [email protected] mailbox, sees old and new messages in it, but cannot write to anyone. Mails are returned with a note: "Failed to deliver to the following recipients or groups:
You cannot send a message on behalf of this user without the appropriate permission. Make sure you're sending on behalf of the correct sender, or request permission if necessary. If the problem occurs again, contact technical support."
I suspect that the problem is somewhere in the rights \ groups of the newly created vasya, because, having done the same with the account of the enterprise admin group, the mail was sent normally. There are some universal means\groups\scripts that can be applied to vasya, so that he can still send mailings to the enterprise?to the developers' recommendation to remove the alias cache, but I think this is a bit off (although I tried removing auto-filled aliases on the client - and this did not change anything). For more information: all the action takes place on win7 pro x64 (ms outlook 2007 mail client) and terminal server MSS 2008 (ms outlook 2010 mail client). Under the admin account, both there and there everything is fine.