I don't understand what you describe here. Break it down step by step.

"Export to an Inbox" just doesn't make any sense to me. Sorry.

What tools did you use to export and import?

Zenos, thanks for highlighting. I meant * .mbox.

So - Export pop mail account to .mbox file from Mac Mail. -deleted pop mail account -Used thunderbird to map to new iMap Gmail for Business account -Used Import/Export tool to IMport .mbox file -Synced

but not history of Sent mail is showing.

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In Thunderbird, Inbox and Sent will each have their own mbox file. Or at least it will expect separate files to import.

I don't know about mac mail and what it offers when you export. But as I understand it, an mbox file represents a single folder. So each of your folders needs an mbox file. Your Sent file may still be there in your profile, or your recycle bin if you deleted it in your file manager.

I can see the Sent mbox files, but it doesn't want to import given the options in the IMport/Export add on tool.