I'm trying to understand the use of projective representations in Clifford theory, and I'd like a small example where projective representations really help, and the ingredients are actually computable.
Curtis–Reiner give an example of $G_1 \times G_2$ which is good in many ways: (1) it shows how the inertia subgroup can be useless in an “easy” case, and (2) how the tensor product decomposition can simplify into something very understandable. However, it has a serious drawback: (3) it can be (and usually is) proven without projective representations, since the end result are all ordinary representations.
Can someone give an example where the projective representations actually matter, and yet where everything is fairly computable? That is, a group (or family of groups) $G$, with normal subgroup $N$ ($N$ not central), an irreducible character $\phi$ of $N$ that is invariant under $G$, and where we can find some interesting irreducible characters $\chi$ of $G$ with $\chi_N = e \phi$ using actual projective representations of $G$.
One of my particular concerns -- It seems that to understand $\chi$ we already need to understand all irreducible projective characters of $G$, so that we must understand not only $\chi$, but all of its distant relatives, before we can understand $\chi$. In $G_1 \times G_2$ some magic simplification happens. I don't see how such magic could ever work again.
I believe the case of semi-direct products (especially Schur-Zassenhaus style) does not qualify, as again all the representations involved are ordinary.