Character triples isomorphism and real characters A character triple is a triple of the form $(G,N,\theta)$ where $G$ is a finite group, $N$ is normal in $G$, $\theta \in Irr(N)$ and $\theta$ is $G$-invariant. For the concept of character triple isomorphism, I refer to Definition $11.23$ of "Character Theory on Finite Groups", Isaacs.
Let be $(\tau,\sigma) \colon (G,N,\theta) \to (\Gamma,M,\phi)$ a character triple isomorphism. Via $(\tau,\sigma)$, some properties are preserved, for example if $\theta$ is afforded by an $\mathbb{F}$-representation and $G$ is perfect, then for every $N\le U\le G$ and $\eta \in Irr(U\mid \theta)$, then $\mathbb{F}(\eta)=\mathbb{F}(\eta^\tau)$ (this appears in an article of Navarro and Tiep of 2009). Unfortunately, for real character the Schur-Frobenius indicator is defenitely an obstacle. If $\theta$ is real, I see no way to make the triple isomorphism send real character in real characters. If $\Gamma$ and $M$ are respectively the Schur covering and the Schur multiplier of $G/N$ and $\Gamma$ is known, then we can get reality provided some uniqueness properties. Namely, if $\Gamma$ has a unique character $\chi$ over $\phi$ of a certain degree, then we can pull the reality to $\chi^{\tau^{-1}}$. The same if $\Gamma$ has an odd number of characters with the same degree, then one of the correspondent character in $G$ are real. These both statements follow easely from the fact that the complex conjugation acts on $Irr(G\mid \theta)$ preserving the degrees. But if none of these conditions are satisfied, I don't know how to proceed. A general tool would be nice.
Just to make it easier, if $\phi=1_M$, then $Irr(\Gamma\mid \phi)=Irr(G/N)$, moreover $\phi^{\tau^{-1}}$ is an extension $\hat \theta$ to $G$ and the map $\tau^{-1}$ is just the map $\gamma\to \hat \phi \gamma$. However, I don't know if $\theta$ is afforded by a real representation, and even if it does, to adapt the proof in Isaacs is kind of difficult. It boils down to the extendiblity of the representation affording $\theta$ or similar criteria (see $11.27$ of Isaacs' book , for example).
I mean, it seems that we are in the context of Gallagher's Theorem, nevertheless, though easier, I can't say much more than this and only some hightlights in this situation would be very useful to me.
 A: I apologize for answering to your question so late. Probably you do not need an answer anymore but I suppose it may still be useful to people facing the same problem.
I also have to apologize for the shameless self-promotion I am going to engage in, since the answer to your question can be found in a paper I recently published.
In fact, the question is answered by Theorem 3.6 of G., Pellegrini, Sylow normalizers and irreducible characters with small cyclotomic field of values (you may also use Theorem 3.5 of the preliminary arXiv version).
In particular, first we have that, if $(\tau, \sigma) : (G,N,\theta) \mapsto (\Gamma, M, \phi)$ is a character triple isomorphism and $\theta$ is real-valued, then also $\phi$ is real-valued. Moreover, if we also have that $G/N$ is a perfect group, then $\chi \in Irr(G \mid \theta)$ is real-valued if and only if $\chi^{\tau} \in Irr(\Gamma \mid \phi)$ is real-valued.
The condition of $G/N$ being perfect cannot be avoided. In fact, suppose that $G$ is a cyclic group of order 4, $N < G$ is a cyclic group of order 2 and $\theta \in Irr(N)$ is the only non-principal irreducible character of $N$. Then, the triples $(G,N,\theta)$ and $(G,N,1_N)$ are isomorphic. However, while all the characters in $Irr(G \mid 1_N)$ are real-valued, none of the characters in $Irr(G \mid \theta)$ is actually real-valued. Notice that, in this case, $\theta$ is afforded by a rational representation (actually, it is a rational representation itself), so I would say that the Schur index does not help in this situation.
