In Game of Thrones as of season 6 episode 7 do the Lannisters know that Little Finger and the Boltons have betrayed them?

Question

Of course WE know that everything Littlefinger does is to his own ends, but Littlefinger went to the Vale on the orders of the Lannisters in the first place. He whisked Sansa Stark away from the Lannisters, and while Sansa maintained her fake identity the Lannisters would have thought Littlefinger still loyal to them. But it's been many episodes since Littlefinger declared Sansa openly in the North, then the Boltons married her to Ramsay Bolton. And everything Roose Bolton and then Ramsay Bolton have done since then indicates that they intend to rule the North without allegiance to the Iron Throne.

But so far I haven't seen any mention of it in King's Landing. They haven't made any moves based on it, or even discussed the fact that that they've lost the Vale and most of the North again along with the allegiance of House Bolton. Is it supposed to be that word still hasn't gotten back to King's Landing about it? Or are the Lannisters conceding the North? Or did I miss something?

Answer 1

Littlefinger assured Cersei that he is still loyal to the Iron Throne. Also, he said that Boltons betrayed the Iron Throne, Ramsay married Sansa. Also, he said that he is ready to send knights of the Vale to storm the North and Winterfell to beat Boltons

Answer 2

Petyr Baelish does as Littlefinger does, he spreads mistruths and information around to cause chaos and profit from it. He placed Sansa in Winterfell to gain the trust of the Bolton's and so he could forge an alliance between the Vale and the North. He then goes to Kings Landing to have a chat with Cersei:

BAELISH: There is another matter, your Grace. Something so urgent I couldn’t trust the words to a raven. You once charged me with finding Arya Stark. To my shame I failed you. But I have found Sansa Stark. Alive, and well, and home again. In Winterfell.
CERSEI: That’s not possible.
BAELISH: My sources are well placed. They tell me Roose Bolton plans to marry her to his son, Ramsay, a bastard recently legitimized by King Tommen.
_{Game of Thrones, S05 E06, "Unbowed, Unbent, Unbroken"}

So it appears that the Lannister's know that the Bolton's have betrayed them though as the information comes from Littlefinger himself they do not know yet that he has also betrayed them.

Note that later on in the conversation Petyr reveals his motives for doing what he has done and also pardons himself later on for when he helps Jon in the Battle of the Bastards:

BAELISH: Perhaps I can help. The Knights of the Vale are some of the best fighters in Westeros, trained to battle in the ice and the snow.
CERSEI: Forgive me Lord Baelish, you have a reputation as a money lender and brothel keeper, not a military man.
BAELISH: You wouldn’t risk a single Lannister soldier or a single coin from the Royal Treasury. What do you have to lose, a brothel keeper?
CERSEI: And if you succeed?
BAELISH: Name me Warden of the North.
_{Game of Thrones, S05 E06, "Unbowed, Unbent, Unbroken"}