In an interview with Jaqen H’ghar (Tom Wlaschiha) with Access Hollywood, he was asked:
Why did Jaqen allowed Arya Stark to leave the house of black and white, his answer was:
In my opinion, it was clear from the beginning. I mean, that’s the reason he trained her. He didn’t train her to become a Faceless Man and stay in the House of Black and White in Braavos forever. I think that Faceless Men, they certainly have an interest [in] what’s going on in Westeros, otherwise why would he [have] just shown up out of the blue, in Season 2 and picked Arya and offered her to train her? So there must be some sort of interest and I think, yeah, what’s the point of training her if she can’t put that training to use?
Was he happy that Arya killed the Waif?
When Jaqen told the Waif, ‘Go ahead and kill her, but don’t let her suffer,’ I think the Faceless Men, in their philosophy, they’re kind of evolved [beyond] all those human things. As I said, death is a part of life, so if Arya had died, Jaqen would have probably regretted it or would have been sorry, but that would have only meant that she wasn’t the person that he’d been looking for.
How did Jaqen change Arya?
When he first met her, he immediately recognized the potential in her. She posed as a boy and everybody but him believed it and so already when she was a little girl, he could see that she had quite some talent to do these things or to become a different person, and when she left Westeros, she was still a little girl and now, over the course of her training, she’s grown up and now she goes back to Westeros and she’s a totally different, yeah, she’s a totally different person. She’s an adult.