Where did Wolverine come from?

Yup – as Bryan Singer and Simon Kinberg have said, Wolverine is the most important part of the X-Men universe. However, his presence in X-Men: Apocalypse leaves quite a few questions unanswered. And the biggest one concerns his capture at the end of Days of Future Past.
As we’ve seen, he’s pulled out of the Potomac at the end of Days of Future Past. And he is captured by Mystique, disguised as Stryker. If this is what happens, how does he end up in the Weapon X program? Mystique has no intention of bringing any harm to her fellow mutants. So having her handing Wolverine over to Stryker is, without a doubt, a false assumption.
The best explanation is that it isn’t Mystique being disguised as Stryker, but some other shape shifter who hasn’t been fully introduced until now (the credits scene for X-Men: Apocalypse is the biggest hint as to whom that is). But, still – this is bothering me, since no genuine explanation is offered in the movie.
It is as if I’m left to believe that for two years that Wolverine is fine and dandy, only to discover much later on that I’m a fool to have believed it. It is as if we were already introduced to the idea of other powerful shape shifters in this universe – in other words, Sinister should have been announced or hinted at earlier for Wolverine’s comeback in the Weapon X program to make any sense.
And speaking of Wolverine and Days of Future Past…
