What Johan said. Here's more detail.
in the "Export location" section of the Export dialog select "Same Folder as original photo" for the "Export to" and check the "Add to This Catalog" and "Add to Stack" check boxes
This will put the exported Jpg into the same folder as the original RAW file, will add it to the catalog and will create a stack of the jpg with the RAW file
In the Metadata section, Select "All Metadata" and uncheck "Remove Person Info" and "Remove Location Info" unless you want to exclude those items from the jpg. It doesn't seem to matter if you check or don't check the "Write Keywords as Lightroom Hierarchy
This will cause all the KW's from the original RAW file to be added to the new JPG
BUT SEE NOTES BELOW
NOTES:
1) Any Keyword that does not have "Include on Export" checked will not be included on the JPG (but its synonyms might)
2) Synonyms where the "Export Synonyms" box is checked but the "Include on Export" box is not checked will come back on the JPG, but as NEW regular Keyword - not a synonym on the original KW
3) A KW that is set to export, but it's parent is not, will come back as a new KW, which is not under the original parent
There may be other permutations I didn't test for as well.
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If you want the exact same keywords as the original you need to take a different approach.
1) Do the export as above but in the Metadata part of the Export Dialog box, select "Copyright Only" instead of "All Metadata"
2) After the import is done, go to the Special Collection "Added by Previous Export", and for
EACH image individually (you can't do this in bulk) do the following:
2a) right click on the Jpg image and select "Go to Folder in Library"
2b) you will now see the original RAW and the new JPG side by side (expand stack if needed)
2c) Select them both, but have the original RAW file as the Active (most selected) of the two
2d) Click "Sync Metadata" at the bottom of the right panel
2e) In the popup, tick the "Keywords" check box. At the same time you should tick or un-tick any other items you want to be synced to the JPG or not Synced to the Jpg such as titles, captions, locations, GPS coordinates, contact info, etc.
2f) Click "Synchronize" at the bottom of the dialog box.
2g) Repeat for each image
Dan