>Knows the faceless men, by "reputation".
CIA never left The Vale...
He hired Jaqen to impersonate him. Jaqen has become obsessed with Arya. Of course Jaqen could out sneak Sansa, he taught her how. Baelish is now a a sneaky, slick ninja? nah... That's also why "Littlefinger" was in tears, because it was Arya of all people, poetically, putting an end to his faceless ways. Notice how "Littlefinger's" love of Sansa increased tenfold after he travelled to the North with Sansa. Baelish briefed Jaqen about his love for Sansa, Jaqen just hammed it up.
Imagine if Jaqen had taken his mask off? I'd would have been extremely painful (for Arya, emotionally)... Also rearrange "for Arya" = For Yara (META MIRROR POTTERY)
Face it Big Guy you're being played here, it's all part of his plan...