Can I publish a paper with a group that is different than the one I start with?
There would be no reason why it would be considered plagiarism—you've done nothing wrong. Getting another group somewhere else to help you get this work published would also not be considered unethical, so long as they actually make meaningful contributions to the paper.

However, if you've done the work all by yourself, why not consider publishing the paper by yourself? Many disciplines have undergraduate journals that would allow you to publish papers without being "sponsored" by a research group.