IMO it woudl be for Emma as the superior in consequence to pay calls on Jane and invite HER to Hartfield not for Jane to push herself forward.

And we're told clearly IMO that Emma IS a bit too conscious of her own superiority, and taht she does not like JAne much. Austen does not say that Jane has ever doen anything to merit this dislike. It may be that Emma IS jealous of Jane's superior accomplishments, she is certainly irritated by teh way Miss Bates goes on about her.. all that is understandable. What is NOT at issue IMO is that she has a duty to be nice to Jane even if she does not much like her.. jsut as she has a duty to be nice to the Bates ladies. I DO sympathise with the irritation at having to call so often when Miss Bates is such a bore.. but really Emma shoudl do it.. even if she grumbles afterwards. but since Jane isn't even in Highbury all that mcuh, it should be easier to make an effort with her.. she may be reserved but she's NOT stupid or a bore.. and given that she has a pretty dreary future ahead of her, it woudl be right for Emma to invite her up to hte house more oftne, invite her ot play the piano or go for walks and drives....
Ti does not matter that she's reserved and not that friendly..