CH holds in V if and only if CH is actually true, for V a model of ZFC2 See Noah Schweber's post on MathOverflow: https://mathoverflow.net/q/78083.
He writes:

Let $V$ be a model of $ZFC_2$. Then I claim CH holds in $V$ if and
  only if $CH$ is actually true. The proof of this is as follows. First,
  $\omega^V$ has order type $\omega$: clearly it has a subset of order
  type $\omega$, and by the second-order version of the powerset axiom,
  $P^V(\omega^V)=P(\omega^V)$, so if $\omega^V$ had the wrong order type
  $V$ would "see" the error. A fortiori, we can deduce that $\omega^V$
  is countable.

Can someone expand on this argument? Specifically I don't understand what relation is implicit in the talk about order types of $\omega^V$ and $P(\omega)^V$, and how the equation $P^V(\omega^V)=P(\omega^V)$ is used in the argument. To be sure: this argument takes place in some first-order set theory (at least strong enough to define $\models_2$ and CH)?
Also, my axiomatization of ZFC2 (see Shapiro's Foundations without Foundationalism) does not include a second-order powerset axiom, only second order replacement: $\forall f \forall x \exists y \forall z (z \in y \leftrightarrow \exists w (w \in x \land z = fw))$. What is the second-order powerset axiom and how does it follow in second-order logic from second-order replacement + the other axioms of ZF?
 A: I'm going to write "$W$" for our assumed set model of $\mathsf{ZFC_2}$ and leave "$V$" (as usual) for the actual universe of sets. Also, I'll conflate $a\in W$ with $\{b\in W: W\models b\in a\}$; this will be especially benign since the first thing we'll show is that $W$ is well-founded and hence isomorphic to some transitive set.

First, let's outline what we need to do.
Think of "CH-in-W" as the statement

$CH_W:\quad$ There is a bijection in W between W's version of $\omega_1$ and $W$'s version of $2^{\omega}$.

(Or more snappily, $W\models (\omega_1)^W\equiv (2^\omega)^W$.)
Meanwhile, "CH-in-reality" (or "CH-in-$V$" if you prefer) is the statement

$CH:\quad$ There is a bijection between $\omega_1$ and $2^{\omega}$.

We want to show (in $V$) that $CH$ is equivalent to $CH_W$. The issue is that we need the relevant objects and collections to "match up:"

*

*$W$ computes $\omega_1$ and $2^{\omega}$ correctly: there are in $V$ bijections $i_1:\omega_1^W\equiv\omega_1$ and $i_2:(2^{\omega})^W\equiv 2^\omega$. An important step towards proving this is to show that $W$ is well-founded, after which we can conflate $W$ with the transitive set it's isomorphic to via Mostowski collapse.


*$W$ computes equicardinality correctly: for $a,b\in W$ such that there is (in $V$) a bijection between $a$ and $b$, there is also a bijection between $a$ and $b$ in $W$. (Note that the converse holds trivially.)

OK, now let me address the second-order powerset bit.
The definition of $\mathsf{ZFC_2}$ that I was taught included the "second-order powerset" axiom; this is basically the statement that true powersets exist in our model. Specifically, it's:

For every $a$ there is some $b$ such that

*

*every element of $b$ is a subset of $a$, and


*for every $X\subseteq a$ there is some $c\in b$ such that $X=c$.

Here capital letters denote second-order variables, and "$X=c$" is an abbreviation for "$\forall d(d\in X\leftrightarrow d\in c)$." Basically, thinking in terms of a structure $W$, we have that $W\models\mathsf{Pow_2}$ iff for every element of $W$ the true powerset of that element also exists in $W$.
This is however redundant: it follows from second-order Replacement. Given $W\models\mathsf{ZFC_2}$, $a\in W$, and $X\subseteq a$, consider the function $F_X:a\rightarrow a$ sending each $x\in X$ to itself and sending each $y\not\in X$ to some fixed $x\in X$. Now apply the first-order powerset axiom inside $W$.
So if you like, you can think of $\mathsf{Pow_2}$ as a lemma rather than a separate axiom.

We are now all set to prove $CH_W\iff CH$. This is a kind of "bootstrapping" argument: we iteratively demonstrate more and more levels of correctness.

*

*First, by $\mathsf{Pow}_2$ we get that whenever $a,b\in W$ we have $a\equiv b\iff W\models a\equiv b$ (think about the set $a\times b$).


*Any linear order in $W$ which $W$ thinks is well-ordered is actually well-ordered. This is because any descending sequence would be an element of $W$ by $\mathsf{Pow_2}$. This implies that $W$ is well-founded (think about the ordinals), and in particular that $\omega^W=\omega$.


*This tells us that anything $W$ thinks is countable is actually countable and vice versa: using $\mathsf{Pow_2}$ we have "countability in $W$" = "in bijection with $\omega^W$ in reality" = "in bijection with $\omega$ in reality" (via the previous bulletpoint). But this is equivalent to $\omega_1^W=\omega_1$.


*Similarly, $\omega=\omega^W$ plus $\mathsf{Pow}_2$ implies $(2^\omega)^W=2^\omega$.
So we get the desired equivalences needed. Continuing this argument we also get e.g. that $W\models 2^{\omega_{17}}=\omega_{18}$ iff in fact $2^{\omega_{17}}=\omega_{18}$; we need to go quite a ways before we get to statements which any $W\models{\mathsf{ZFC_2}}$ could be wrong about.

Finally, as you say this all takes place inside a theory strong enough to talk about second-order logic over set-sized structures. The usual first-order theory $\mathsf{ZFC}$ is indeed sufficient for this task. Specifically, just like first-order logic over a set-sized structure $A$ is dealt with at the level of $\mathcal{P}(A)$, second-order logic over a set-sized structure $A$ is dealt with at the level of $\mathcal{P}(\mathcal{P}(A))$. So $$W\models\mathsf{ZFC_2}\implies(CH\iff CH_W)$$ is formalized in the language of set theory as a statement of the form $$\forall w\mbox{[stuff about $\mathcal{P}(\mathcal{P}(W))$]}\implies(CH\iff \mbox{[stuff about $\mathcal{P}(W)$]}).$$
Note that the above means that what we're really proving is that every set-sized model of $\mathsf{ZFC_2}$ is correct about $CH$. This is because of a limitation of $\mathsf{ZFC}$: it can't even talk about the satisfaction of first-order theories in proper-class-sized structures, let alone second-order theories in such (unless those theories are of bounded quantifier complexity - which ours aren't). If we want to treat class-sized structures we need to pass to a hyperclass theory (just like we need to pass to a class theory in order to talk about first-order  semantics of class-sized structures).
A: Second-order power set axiom would denote the following statement: if $a$ is a set and $P(x)$ is any second-order predicate which satisfies $\forall x (P(x)\to x\in a)$, then there is $b$ such that $b=\{x\mid P(x)\}$. This follows from the full second-order separation: take $b:=\{x\in a\mid P(x)\}$. Since the second-order replacement proves second-order separation, we are done.
$V$ has its 'own' power set operation, since it is a model of $\mathsf{ZF}_2$ (hence that of $\mathsf{ZF}$.) Noah Schweber's argument shows the internal power set $P^V(\omega^V)$ of $\omega^V$ in $V$ coincides with the true power set $P(\omega^V)$ of $\omega^V$: since the second-order power set axiom (under the full semantics, of course!) catches arbitrary subsets of $\omega^V$.
If $\omega^V$ is not standard (i.e. $\omega^V\neq \omega$,) then $\omega$ is a proper subset of $\omega^V$. Since $V$ contains all subsets of $\omega^V$, we have $\omega\in V$. We can see that $\omega$ is the least inductive set in $V$, but it contradicts with the definition of $\omega^V$ (the least inductive set in $V$.) This shows $\omega=\omega^V$.
As you mentioned, this argument works over a set theory that can formulate $\models_2$. Especially, this argument works over $\mathsf{ZF}$. To note, however, the existence of models of $\mathsf{ZF_2}$ is not provable from $\mathsf{ZF}$ alone. Its existence is equivalent to the existence of an inaccessible cardinal. Our argument would be nothing more than meaningless if we have no models of $\mathsf{ZF_2}$, although the preceding argument holds.
