First, a man who helps Arya in season 2 is not necessarily also a man training her in season 5. A man may have many faces, and a face of no-one may appear on many men. It is known.
In the books, that particular no-one has a very different trajectory to Braavos, and is almost certainly last seen (mystery spoiler in AFFC)...
impersonating a dead Citadel acolyte, "befriending" Sam, having killed a seemingly innocent young man to acquire an Archmaester's key
Whatever a man intends to do, it appears more complex than simply helping a girl become no-one.
And there's yet another layer to the mystery. A few things we learn about those dungeon black cells in the books, mostly in AFFC:
- The man in charge of them (therefore, in charge of keeping Jaqen imprisoned), Rugen, was a mysterious character, "no friends, no kin", "The other turnkeys were afraid of him", "was seldom here", "came and went as he pleased". He appeared under mad Aerys (not long after Aerys hired Varys), and disappeared after Tyrion's escape (leaving behind a coin that caused Cersei to suspect the Tyrells were behind this)
- Jaime, while pretending to "investigate" Tyrion's "mysterious" disappearance, appears to know some secret about who he really is:
"Rugen" the old man supplied. "An undergaoler. He had charge of the
third level, the black cells."
"Tell me of him," Jaime had to say. A
bloody farce. He knew who Rugen was, even if [this old man] did not.
- Very few people were kept in the cells (quote from the above old man):
...we had Grand Maester Pycelle for a time, and before him Lord Stark the
traitor. There were three others, common men, but Lord Stark gave them
to the Night's Watch. I did not think it good to free those three, but
the papers were in proper order
- Those three are clearly Jaqen, Rorge and Biter - and the timing of that last point doesn't quite add up. Ned was dismissed the day after Yoren arrived late at night looking for "scum" to recruit from the dungeons. Doing the paperwork over breakfast that morning doesn't seem right, and nor does doing it after his reinstatement. Even if Ned had had time to, he'd have been unlikely to judge Rorge and Biter worthy of becoming brothers to his brother. Did someone else (Rugen perhaps?) forge the papers sending these three dangerous men north?
- Rugen's description also precisely matches that of the "grizzled", stubbly gaoler Varys disguises himself as in AGOT when visiting Ned in the black cells...
Yeah, we can safely assume that Rugen is one of Varys's disguises.
This means Varys was in control of the black cells.
This suggests, Jaqen was essentially Varys's prisoner - or his guest.
Alongside a cannibal and a twisted sadist. An interesting selection of guests...
It makes sense that a master assassin making investigations would cross paths with that city's master spy - and whether they decide to be enemies or allies, they will want to keep a close eye on each other.
I've also always thought that a black cell with an unlocked door and a cooperative master spy jailer, connected to a secret network of tunnels, would be a convenient base for an assassin to conduct investigations. And it'd explain why Rorge and Biter were terrified of him, if he habitually escapes, then comes back.
Varys would have been able to prevent Yoren taking them, had he wanted to, and may have gone as far as forging the paperwork to have Yoren take them north.
Why? A user does not know... but it adds an interesting dimension to the mystery.