Strangely, the rumours about Serena being heartless bully seems to be spreading through the servants gossip network.
Both are supposed to have had their social debut during the tea party. Even if Serena took after her mother and prefer not going to social event, Rosemary is the type who would love to gossip and brag about herself, she would be excited to go to social parties. Her status as a prince's fiance would also bring people who try to use her to build connection.
In those case, it's likely Serena's mom will asks her to tag along (and she need to keep an eye so Rosemary don't do anything extremely stupid).
But so far, it doesn't seem like they're doing socializing at all.