No Sent Mail Box on my iPhone 4s

I thought I had resolved this problem, but I was wrong. See thread of May 8, 2012 in this section.

This problem began in the spring of 2012. I use Google Apps IMAP. On my iPhone my Sent Mail Box disappears after a time (sometimes a day, at other time a week or more). Before upgrading to iOS 6 I was able to restore my Sent Mail Box by deleteing my Mail account on my iPhone and recreating it. Again it lasts only a few days. I thought I had resolved this problem by unchecking "Store sent messages on the server" on the Mailbox Behaviors tab in the Accounts section of Mail preferences on my Mac, but I was wrong.

Now that I have installed iOS 6 on my iPhone 4s, I am unable to create a Sent Mailbox on my iPhone for even a short periord of time. All my other Mail Boxes/Folders on Google Apps Gmail appear on my iPhone except the Sent Mail. My Sent Mailbox appears on my iMac and my Macbook Air.

I have the same problem with my iPad (first generation). However in my iPad, on which I am still running iOS 5, I can create a Sent Mailbox but it lasts only a few days. Again, on my iOS 6 iPhone 4s I can not create a Sent Mailbox.