But Caroline deliberately misled Jane regarding their friendship, to the point of being actively cruel to Jane, and it's not actually clear whether she's in love with Darcy or just a social climber. Her attitude towards Jane is actually indicative of the latter--it's not just that she wants her brother to marry Georgiana Darcy b/c it'll give her an in with the brother, but that in doing so, Bingley will further establish the family as upper-crust gentry, rather than new money. Her desire to retain rights to visit Pemberley, as Austen phrases it, also seems to indicate social climbing, as the language privileges Pemberley above either of the Darcys in Caroline's desires.

She doesn't treat Elizabeth well--she "paid off every arrear of civility to Elizabeth." So she's never rude, but she's clearly not ever truly warm. It's a very dutiful line.