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P&P
When breakfast was over, they were joined by the sisters, and Elizabeth began to like them herself, when she saw how much affection and solicitude they shewed for Jane.
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The sisters, on hearing this, repeated three or four times how much they were grieved, how shocking it was to have a bad cold, and how excessively they disliked being ill themselves, and then thought no more of the matter; and their indifference towards Jane, when not immediately before them, restored Elizabeth to the enjoyment of all her original dislike.
Elizabeth is perceptive but the Bingley's sister made a lot of Jane and sought her friendship. Even E. was fooled by their behaviour when Jane was present. They made no real effort to hide their true personalities from E.
I think Jane believed her friends would welcome her to their home and perhaps she secretly hoped to meet their brother again. If Darcy did know perhaps he thought she was there at the behest of her mother.