This post contains spoilers from The Diplomat season 3 episode 3 from this point forward.
After The Diplomat season 3 began with a rapid presidential succession and Penn's unexpected appointment of Hal as her vice president, Kate was left in the middle and decided to remain in London as an ambassador rather than go with Hal to DC as special envoy to Europe. Picking up after that shocking decision, season 3 episode 3 opens with Kate touching down at the White House to attend a "gathering."
Back in London, Eidra's assured that there were no red flags in the coroner's report of Roylin's death, but she still needs to find a way to save her job. The Wylers are also asked to make their marriage and respective jobs work as Billie speaks with them about the language they use to describe their lives. She wants them to express more than logistics and express their love for each other.
When they're alone, Kate and Hal argue about keeping up appearances. A private divorce is still on the table. Something suggests that the Wylers won't be about to split as easily as they think they can, but we'll see what happens. Stuart suggests that Eidra calls on Kate's connections with the director of the CIA as the Second Lady to help her out. Kate discovers that Penn appears to be considering dumping Hal and picking Synar, the governor of Pennsylvania, for VP instead.

President Penn considers dumping Hal as vice president
Kate vows to Billie that she will rein in Hal and make sure that he gets the vice president nomination from Penn. When Hal pushes back against Kate's instruction, she admits that he's on thin ice with Penn. Hal admits to Kate that he's three votes away from passing the Law of the Sea Treaty. He's been doing it behind Penn's back in case it doesn't happen, but if it does, Penn will look great. Kate's in on helping Hal with working to get the last three votes.
Stuart learns that Kate missed the meeting with the head of the CIA, which isn't received well and sets Eidra back. Billie dupes Kate and Hal into having dinner together since she's frustrated that their arrangement isn't working. Kate insists that Hal loop in Billie on the treaty to prevent her swaying toward Synar. Kate doesn't end up sitting down for dinner, and on the way to Rayburn's funeral, she leaves a message for Diane at the CIA.
When Hal spots Synar putting in face time with Penn at the funeral, Kate insists that he steps in to make himself known and seen. Synar tells Penn they need to "look behind Pennsylvania," and even though Kate tries to keep him from opening his mouth, Hal rebuts Synar's point. Penn appears to be both impressed and worried about Hal stepping in the way that he does.

Billie attempts to advocate for Hal to Penn, but she compares him to cilantro. Perhaps she wasn't as impressed with him as we thought. Penn asks to talk with Kate and admits that she doesn't have a VP nominee because Hal can't control himself and Synar isn't competent. Hal insists to Billie that Penn needs to move on from fulfilling Rayburn's agenda and make her own moves.
Penn wonders if Hal can be trained, but Kate tells her that she can't promise Hal won't bark at a political rival in public again. She decides to reveal Hal's progress on the Law of the Sea Treaty as a Hail Mary. Kate's plea appeared to work. Penn will take a chance on Hal as vice president with plans to talk strategy on the way back to DC. On the way out, Kate and Hal hug and play up their relationship for the nearby photographers.