In the teaser for the upcoming fourth season of the show, set to hit the streamer on 15 November (20), the monarch is seen looking forward to having a female prime minister.
As Tobias Menzies' Prince Philip says, "Two women running the show - that's the last thing this country needs," the Queen responds, "Perhaps that's exactly what this country needs."
"I think we have enough respect for one another personally to ask ourselves some of the bigger questions, woman to woman," Anderson's Thatcher is heard saying in a voiceover narration.
Writer Peter Morgan muses in a press release, "It could be said that they are two very similar women, born only months apart... It was a commonly held piece of wisdom that the Queen and Thatcher didn't get on."
"I think there was probably a lot of respect. But they also had many differences, it was really fun exploring their differences and their similarities"
Meanwhile, Prince Charles, played by Josh O'Connor, is tasked with finding a woman who "people will love as a princess, and in due course, as queen," with the clip offering a glimpse of Emma Corrin as Princess Diana rollerskating and dancing around the palace, as well as being swarmed by admirers and paparazzi.
Princess Margaret - portrayed by Helena Bonham Carter - asks, "How many times will this family make the same mistakes, paying the consequences each time?"
The show will follow the royals as the 1970s ends and Britain is divided by Thatcher's policies - with Prince Charles' romance with Diana providing a welcome distraction for both the public, and the Queen, as she looks to her heir to secure the line of succession to the throne.