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I think you're thinking of Juliet (not Kate)...Sawyer, Juliet, Miles, Jin, Daniel Faraday (along with Rose and Bernard) were transported thru time, thru various time periods...eventually ending up stuck in 1974-1977. A few of the survivors were eventually welcomed into "The Dharma Initiative" others stayed hidden outta site- Rose and Bernard. It's unclear if anyone elese survived both the Freighter explosion and the Time Jumping phase, and later the shower of light up fire arrows shot by "The Others (in 1974)"

There's also some time, after the Ajira plane arrived at the island that Kate, Jack and Hurley were brought in by Sawyer to the "Dharmaville" as new recruiters. After a while Sayid was also captured by Dharma security people as a terrorist.

And also, when the Oceanic 815 survivers were divided into two according to those who want to follow Jack and trust the freighter people and leave the island; or those who followed John Locke not trusting freighter people and Jack, or not wanting to leave the island. That's when Kate first went with Jack and then, went to live in "Dharmaville" with Locke and Sawyer. But that was for one or two nights until Locke tried to banish Kate for trying to speak to Ben Linus without asking for John's permission.

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