nt.number theory - obstruction to taking the square root of a Galois character

With luck, this will be blunder-free (and if it's not, please tell me!):

Consider the short exact sequence
$$1 to mu_2 to overline{mathbb Q}^{times} to overline{mathbb Q}^{times}
to 1,$$
with the third arrow being squaring, and with trivial $G_k$-action. Passing to cohomology,
the sequence of $H^0$s is just this same sequence, and the sequence of higher cohomology
becomes
$$0 to Hom(G_K,mu_2) to Hom(G_K,overline{mathbb Q}^{times}) to
Hom(G_K,overline{mathbb Q}^{times}) to H^2(G_K,mu_2),$$
with the last arrow being the obstruction you asked about.

Added: See Brian Conrad's comment below for a cleaner point of view, showing
that $H^2(G_K,mu_2)$ is the precise obstruction space.