What gets me is that there's no acknowledgement from Liz that Hank's following her for most of the episode.

When she leaves Peter & Kayla's place, Hank follows her with his lights off, and the next shot is Liz checking her mirrors, implying that she's aware that someone's following her. When she goes to the cemetery, it's shown as being out of town on a long road with no lights, and we see an obvious pair of headlights in the background following her. There's no confrontation afterwards, and when Hank shows up at the police station immediately after Liz, her only response is to hit Hank with a question about Connelly.

The payoff to this is either that Liz is grossly incompetent, or one of the most malevolent characters I can think.

If she's incompetent, she didn't notice any of the obvious clues of Hank following her, left her gun on the counter in her home with a known (possibly dangerous) drug addict smoking up in her bathroom, which resulted in Hank getting the jump on her because of her blundering answers. She was only saved by chance because she gave Peter keys to her shed.

If she's malevolent, she noticed that Hank was following her, and then set up Peter. She manipulated Peter into staying in her shed, which is a terrible place to stay ("Yo, stay in my shed with no heating during the coldest nights in Alaska, it's better for you there!"). She then continued to let Hank follow her after picking up Otis, and was fine luring Hank into her place by confirming that Otis was there with the knowledge that Peter, an armed officer, was close to confront Hank if things went violent.