I use Thunderbird 3 with my GMail account using IMAP. When I send messages via Thunderbird to Google Groups I am subscribed to, my own messages do no appear in threads. Looking at the same threads in the GMail UI and the Groups UI, my own messages show up just fine.
Is there a way to get my own messages to show up in Thunderbird?
Gmail de-duplicates emails. When you send a mail, the copy which comes back in is found to match the copy you already have and ignored. This is why mail you send from your own address often just appears in Sent Mail.
Sent Mail is still in the conversation, and should still have the References: headers to allow it to be threaded.
I suspect that you're using Gmail filters to tag mails based on the list as they come in? So this works in Gmail because when you look at a label in Gmail, you select conversations (threads) rather than messages, so you get all the replies. The mail you sent won't itself have the label though, so on a strictly message-based interface such as IMAP that mail won't appear in the folder.
If you're using listid: to filter mail in Gmail, try using list: instead. The list: operator handles the To: header, as well as List-Id:, so is better than the listid: operator (which is just too low-level). Use <...> around the list-id to change the tokenisation to be more appropriate (and the '@' is a magic @ here, which will match the '.' form in the List-Id: header).
I have just solved this problem using the Send Filter add-on for Thunderbird.
Note, a TB3 compatible version is available.
Send filtering is also supposedly going to be available natively in TB3, but as of the 3.0 release, I can't see that it's in there.
With this add-on installed I am simply creating filters on the To: field for the relevant lists and setting the action to copy the message to the relevant folder. Works just fine.