Question about preimage of conjugate class of quotient group $G$ is a group, $K \triangleleft G,\ |K|=2,\ \overline G=G/K$. 
Suppose $\overline C$ is a conjugate class of $\overline G$, $S$ is preimage of $\overline C$ in $G$. Then one of these two holds:
(1) $S=C$ is a single conjugate class with $|C|=2|\overline C|$.
(2) $S=C_1 \cup C_2$ is union of two conjugate classes with $|C_1|=|C_2|=|\overline C|$.
My problem:
What does "$\overline C$ is a conjugate class of $\ \overline G$" mean?
Since conjugate class of $\overline G$ in $\overline G$ is just $\overline G$ itself, I'm not sure what this $\overline C$ refers to.

$\bar{C}$ being a conjugacy class of $\bar{G}$ just means, that $\bar{C}\in\bar{G}$, i.e. $\bar{C}=g\circ K$ for some $g\in G$. The preimage then is:
$$\{g\in G: g\circ K=\bar{C}\}$$
Elements of a quotient structure are almost always called conjugacy classes, as they are defined via an equivalence relation :)
